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 + iooL, Y = G + io^C. (3) 



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

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

 suggested in (3). 



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

 solutions : 



where 



T = a + ip = 4ZY, ^0 = Vf "^ ^ " T ' 



(4) 



It is customary to designate by V 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 T and Zo are called, respectively, 

 the propagation constant and the characteristic impedance. The real 

 part oc of the propagation constant is the attenuation constant and j3 

 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 -f CO = 0, c = ^ . 



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

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

 in the opposite direction. 



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

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

 points is called the wave-length. By definition 



^j8X = 27r, ^ =^- 



