IMPEDANCE CONCEPT AND APPLICATION 21 



The coefficients of proportionality Z and Y are known as the dis- 

 tributed series impedance and the distributed shunt admittance of the 

 Hne ; they depend upon the distributed series resistance R, shunt con- 

 ductance G, series inductance L and shunt capacity C in the following 

 manner : 



Z = R + ic^L, F = G + icoC. (3) 



In a generalized transmission line Z and Y may be functions of x 

 and may depend upon w in a more complicated manner than that 

 suggested in (3). 



If Z and Y are independent of x, (2) possesses two exponential 

 solutions: 



/+ = ^g-r-+-', V+ = Zo/+; 



(4) 



where 



r = a + i|S = VZF, 



\z _v _ 



-\Y~Y~ 



It is customary to designate by T that value of the square root which 

 is in the first quadrant of the complex plane or on its boundaries ; the 

 other value of the square root is — F. 



The two "secondary" constants F and Zo are called, respectively, 

 the propagation constant and the characteristic impedance. The real 

 part a of the propagation constant is the attenuation constant and j8 

 is the phase constant. 



Equations (4) represent progressive waves because an observer 

 moving along the line with a certain finite velocity beholds an un- 

 changing phase of V and /. This velocity c is called the phase velocity 

 of the wave. Setting x = ct m the upper pair of (4), we obtain the 

 condition for the stationary phase 



-/3c + co = 0, ^=|- 



Hence, F+ and /+ represent a wave traveling in the positive x-direc- 

 tion. Similarly we find that V~ and I~ represent a wave traveling 

 in the opposite direction. 



Consider two points in which the phases of V and / differ by Iw 

 when observed at the same instant; the distance X between these 

 points is called the wave-length. By definition 



j8X = 27r, X =^. 



