STEADILY ALTERNATING CURRENT ON A LONG LINE. 355 



of i.o Z 0° volt max. cyclic value. After this wave of e.m.f. has 

 reached P, it will have attenuated to the vector perunitage 



^-0' ^ ^-ie.'+je,') ^ ^-e,' ^ ^-je.j ^ ^-e,' ^ ^^j numeric Z . (21) 



where d^' is expressed in real circular radians (or imaginary hyper- 

 bolic radians), lagging in phase behind the generator e.m.f. at that 

 moment. When the wave has reached B, the receiving end of the 

 line, its condition will be e' =e~^^\~ On. At the junction BC, the 

 wave breaks into a transmitted and reflected component. If the 

 transmission coefficient is nv, then, following in planevector nota- 

 tion what Heaviside^- first showed, with real numbers only, for the 

 distortionless case : 



m = j — = . — numeric Z , (22) 



and the e.m.f. transmitted to o- at C is me ^ volts Z. (23) 



The voltage for reflection at B will be (i — w)"* volts. This 

 voltage is directed back from B towards A, and should be counted 

 as — (i — ni)e~^ = (w — i)e~^ volts Z on the line. 



The time required for the wave to reach P from A will be 6o'/o> 

 seconds, and to reach B, OJin seconds. This is the traverse time of 

 the line and may be denoted by T. We may therefore represent the 

 progress of the wave by the following table. The assumption is 

 made as a first approximation, that the generator at A has negligible 

 internal impedance ; i.e., that the effect of its internal e.m.f., acting 

 in conjunction with reflections from its actual internal impedance, 

 is the same as its terminal e.m.f. steadily impressed at A would have 

 with zero internal impedance (Table III.). Here S^ is the position 

 angle at the end B of the line in the steady state or tanh "^(cr/^'o). 

 Again 8^=<9-[-8i; is the position angle at the end A of the hne in 

 the steady state. Also 8p = (9" + 8^ = 8 .j — ff is the position angle 

 at the selected point P of the line in the steady state. It is well 

 known that, in the steady state, if the impressed voltage at A is 

 1.0 Z 0° max. cy. volts, the final voltage at P is (sinh Sp)/(sinh 8^). 



12 Bibliography i. 



