IMPEDANCE CONCEPT AND APPLICATION 33 



the terminal impedance by k, we use (15) to rewrite (14) in the follow- 

 ing form : 



li -\- Ir = It, li — Ir = -T, k = -^ . 



Solving, we obtain the reflection and transmission coefficients : 



(16) 

 r = -' = 2^ T =Y^= ^ 



' li \+k' "" F. \+k- 



Thus when k — \, that is when the terminal impedance equals the char- 

 acteristic impedance, there is no reflection. When the ratio of the imped- 

 ances is zero or infinity the reflection is complete: in the first case the 

 current vanishes and the voltage is doubled, and in the second the current 

 is doubled and the voltage vanishes. The amount of reflection is com- 

 pletely determined by the ratio of the impedances . 



The terminal impedance may be another semi-infinite transmission 

 line and its characteristic impedance will play the part of Zt. It is 

 important to note that neither the propagation constants nor the velocities 

 of the wave in the lines have anything to do with reflection. No reflection 

 will take place if the lines have equal impedances and there will be 

 reflection in the case of unequal impedances even if the velocities are 

 the same. 



The variables V and / can stand for any two physical quantities 

 satisfying equations (2). It will be observed that if we disregard the 

 physical significance of the variables V and /, the characteristic im- 

 pedance can be defined either as the ratio Vjl or as IjV. We are 

 perfectly free to make our choice. It is evident from (16) that if we 

 interchange V and / and replace k by its reciprocal, the expressions 

 for the reflection and transmission coefficients remain unaltered. 



Non-Uniform Transmission Lines 



The foregoing analysis has to do only with uniform lines. In the 

 case of non-uniform lines the impedances looking in the opposite direc- 

 tions may be different. These two impedances will be defined by the 

 following equations : 



_V+ _ _ V- 



^0^ = -J+ J Zq = — yz: , 



where F+, /+ and V'^, I~ refer to the two waves. 



