458 BELL SYSTEM TECHNICAL JOURNAL 



^^''' -' = KTTKb (27) 



Vh = Vf+V„ = 2^'=^^^ Va(l + r)(l - r) (28) 



V, = e-<P^V, = ze-''^'V.{l + r){-r) (29) 



F„. - zV, = z'^e-''^'^ Va(l + /•)(-;-) (30) 



F„ == e-'"^^7„. = sV'^^ F.(l + r){-r) (31) 



Kp = -rVn = 2^r''^^F„(l + r)(-r)'^ (32) 



V, = V„+ Vp= 2V'*^^Fa(l - 0(-r) (33) 



For purposes of analysis it is now assumed that further multiple reflections 

 are negligible. 



13? 



Fig. 11 — Vector voltage diagram for maximum vector sum. 



3 2 1 



* « ^ 



Fig. 12 — Vector voltage diagram for minimum vector sum. 



Equations (19), (28) and (33) are the reflected voltages that combine 

 vectorially to be measured. If ^L = 0, 7r, 2x , ■ • • nw then the vector 



voltage diagram might appear as in Fig. 11. If BL =-, — , — , • • • 



— then the vector voltage diagram might appear as in Fig. 12. 



The followmg example illustrates the calculations involved in computing 

 the errors due to the magnitude of the reflection coefiicient being measured. 

 The assumptions are such that an appreciable error is computed. If one 

 assumes r = 0.316 and z = 0.282, then from equation (6) TIV = 10 db 

 and T^. = 11 db. In Figs. 11 and 12, 



vector 0-1 = r, vector 1-2 = z(l — r'-), vector 2-3 = rs-(l — r-) (34) 



then 



TFo_i = 10 db, IFi-2 = 11.00 + 0.92 = 11.92 db, 



and IFo-s = 10.00 +22.00 + 0.92 = 32.92 db (35) 



In order to evaluate vector 0-2 in Fig. 11 (the vector sum of vectors 0-1 

 and 1-2), one calculates their difference T. 



T = 11.92 - 10.00 = 1.92 db (36) 



For T = 1.92 db, /'i = 5.10 db (37) 



therefore W0-2 = 10.00 - 5.10 = 4.90 db (38) 



