WAVE PROPAGATION OVER CONTINUOUSLY LOADED WIRES 203 

 determinant (22). Therefore 



w 



here 



A21 = KiAn, 



A 22 ^^ -'^2-'4i2i 



(32) 



^12 — ^2*2 — T r~ ^22 — —■^22 — 



p 2 p 2 



ZT~ ^22 — —^22 — ^, 



Z22' 7 , r,^ 



- Z.2 + 3r 



F - F.- 



Z7 ' ~ 7 7' - 



{2^2>) 



Zii ^ . r 



2 



-Z22+Y: 



Substituting (32) in (31) and solving 



A - T ^^ A - - T ^' 



-^11 — -' ^? T^ } ^12 — -f 



-A.2 — -A-i 7V2 — iV 1 



^21 — -'O'i? ^ — ~ ^22- 



A2 — Ai 



(34) 



Finally, the currents are given by 



/n 





(35) 



This completes the analysis for the more general system where 

 the magnetic sheaths are insulated from the wires. For the special 

 case where wire and sheath are contiguous, 712" is infinite and (28) 

 shows that Fi = 7 and r2 = «> . The transmission is, therefore, 

 defined by only one mode of propagation. The series impedance of 

 the system is, from (23) and (26), 



Z - 2 |^Z22" - z,- - Zo/ + Z22' J + ^'' 



(36) 



where the terms in brackets give the internal impedance of one of 



the loaded wires, and X2 is the reactance that arises from the magnetic 



field between them. The internal impedance can be obtained also by 



2£./' 

 finding -j — f-f- directlv from the last two of equations (1), of course. 

 l\ ~T Li 



