IMPEDANCE CONCEPT AND APPLICATION 35 



toward P. The effect of the inserted piece is comprised of two inde- 

 pendent reflections at P and Q and of attenuation with concomitant 

 phase change between P and Q. Thus the transmission coefficients 

 across PQ, that is, the ratios of the quantities at Q to the impressed 

 quantities at P, are 



Ti = Tr, p Ti, Q e ^"\ Tv = Ty, p Tv, q e' 



V"l 



where Tt,p is the transmission coefficient for / at P and the remain- 

 ing 7"'s have similar meanings. 



If PQ is a piece of a uniform transmission Hne inserted into a uniform 

 semi-infinite line, Zt = Zo'. In this case, we have 



Ti = Tv = 



where k is the ratio of the characteristic impedances. The factor 

 4:k/(k + 1)2 represents the reflection loss and e~""' the attenuation 

 loss. 



Let us now assume that PQ is electrically short and that all the 

 transmission lines in question are uniform. By the transmission line 

 theory, the ratios of the total currents and voltages at P and Q are : 



lo Z(] 



Ip Zo" cosh r'7 -\- Zt sinh T"l ' 



Vq ^_ Zt 



Vp Zt cosh r'7 + Zo" sinh T"l ' 



On the other hand, we have 



Ip 2Zo' Vp 2Zp 



li Zo'-^Zp' Vi Zo'+Zp' 



where Zp is the impedance at P looking toward Q 



y ^ y „ Zt cosh V"l + Zo" sinh V"l 

 '' ~ ° Zo" cosh V"l -f Zt sinh V"l ' 



The transmission coefficients across PQ can be represented as 



^ _Iq _Iq Ip ^ _Vq _Vq Vp 



^'~h-TpTr ^'-y^-v-pTr 



Making appropriate substitutions into this equation, we obtain 



Tr = pil - qe-^^"T'e-^"\ Tv = ^, Tj, (17) 



