32 BELL SYSTEM TECHNICAL JOURNAL 



when the electromotive force driving the current is distributed uni- 

 formly along the filament. In this case we have 



^ _ r}IKo{(Tp) ^ IKi(ap) ^ Kojap) 



2TaKi(<ra)' " 2TraKi{aa); ' '^Ki(ap)' 



where I is the current in the filament. On the other hand if the 

 electromotive force is applied to the filament with a uniform progres- 

 sive phase delay so that it varies along the filament as e~'''" for in- 

 stance, then the field and the impedance are 



^ vIK,{Tp) _ IK^jTp) 



PART II 



REFLECTION, REFRACTION, SHIELDING AND POWER 

 ABSORPTION— GENERAL FORMULAE 



Uniform Transmission Lines 



While the following discussion refers specifically to an electric trans- 

 mission line, the results apply to all generalized transmission lines of 

 which the former may be considered typical. These results depend 

 upon certain boundary conditions and are not influenced by the names 

 of the variables. 



Consider a semi-infinite transmission line terminated by a pre- 

 scribed impedance Zt. Suppose that an "impressed" wave is coming 

 from infinity. If F, and /»• are the voltage and the current, their ratio 

 must equal the characteristic impedance Zo of the wave. On the 

 other hand, the ratio of the voltage across the impedance Zt to the 

 current through it is Zt by definition. Thus, unless Zt is equal to Zo, a 

 "reflected" wave must originate at the terminal and travel backwards. 

 Let Vr and Ir be the voltage and the current of the reflected wave at 

 the terminal. The total values of the voltage and the current will be 

 designated by Vt and It. Then at the terminal 



Ii + Ir = It, F. + Vr = Vt. (14) 



By (9) and by the definition of Zt, we have 



Vi = Zoli, Vr= - Zoir, Vt = Ztlf (15) 



Designating the ratio of the characteristic impedance of the line to 



