WIRE TRANSMISSION THEORY 269 



where Z is the uniformly distributed series impedance and Y the 

 shunt admittance per unit length, it is easy to show that I and V 

 satisfy the differential equations 





y'-^.)v=o, 



1=0, 



(3) 



where y^ = ZY. The solution of these equations is 



I = Ae--^' - Be"', 



(4) 



V = kAe-^' + kBe^'. 



7 = VZF is called the propagation constant and k = -{zJY the 

 characteristic impedance of the line. A and B are integration con- 

 stants which must be so chosen as to satisfy the boundary conditions 

 (continuity of current and potential at the line terminals). The first 

 term represents a direct wave, the second a reflected wave, their rela- 

 tive values depending on terminal reflections and the terminal im- 

 pressed electromotive forces. 



We see therefore that in accordance with elementary or classical 

 transmission theory, the current and potential waves are both express- 

 ible as unique simple exponentially propagated direct and reflected 

 waves, the values of which are determined by the continuity of current 

 and potential at the line terminals. The characteristics of the line 

 appear only through two parameters, the propagation constant 7 and 

 the characteristic impedance k. 



Generalizing the preceding, consider a system of n parallel wires, 

 parallel to the surface of the earth. The differential equations for 

 such a system, in terms of elementary transmission theory, are ^ 



tz^^h = -j-^Vi U= 1, 2.-.«), 



f:Y,,V,= -^li {j= 1, 2-..n). 

 t=i ox 



(5) 



Here the physical system is represented by the parameters Zjk and 

 Yjk, the Z parameters being the series impedances (self and mutual) 

 and the Y parameters the shunt admittances. If the differential 

 operator djdx is replaced by 7, thus confining attention to exponentially 

 propagated waves, and if either the potential V or the current / is 

 * See references 9 and 10. 



