WA VE PROP A GA TION VER CONTIN UO USL Y LOA DED WIRES 1 9 1 



what the effect would be of introducing insulation between the wire 

 and its sheath. In this more general system, shown by the sketch, 

 the solution for the propagation constant has two values, because 

 two layers of insulation are involved, and cannot be expressed in 

 the usual form. It is found, however, that it can be expressed in 

 terms of the propagation constant for the elementary case where wire 

 and sheath are in contact by introducing two other known propagation 

 constants that determine transmission along the separate pairs of 

 conductors in the system. The expression for the propagation con- 

 stant, when given in this form, shows directly the effect of insulating 

 the wires from their sheaths. 



It is necessary first to define certain impedances. Let /] be the 

 total current in one of the wires and lo the total current in its sheath. 

 The tangential electric forces in the surfaces of wire and sheath are 

 denoted by Ei" , £«' and E^" , as shown in Fig. 1. These electric 

 forces are linear functions of the currents, as follows: 



-C,2 — -^21 1 1 I ■'^22 -'2) 

 -C,2 ^^ •^21 J 1 ~r ^^22 ■*2) 



E," = Zn"h. 



(1) 



The impedances which appear in these equations as the coefihcients 

 of the currents are functions of the electrical constants and dimensions 

 of the wires and sheaths. Their values are given in Appendix A. 

 Now let 



7 = propagation constant determining transmission along the loaded 

 wires if the wires and their sheaths were in contact = VKoZ. 



7i2 = propagation constant determining transmission along one wire 

 with its sheath as the return, when the sheath is insulated 

 from the wire = VFiZio. 



722 = propagation constant determining transmission along the two 

 sheaths if the wires were removed = V ^2^22. 



Then, from (1) 



£2' - Ey" 



Z12 — 



Z22 = 

 Z = 



I Ai — Zii — Z21 ~l~ Z22 -(- Ai, 1-2 = — /i 



2E2" 



h 



IE." 



+ A2 = 2Z22" + X: 



In these equations, 



+ X2 = 2 



Z22 



7 'i 



Z]i — Z21 ~r Z22 



/i = y (2) 



+ X2 



