TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 111 



series resistance i?„ and inductance L„, and distributed shunt capacity 

 Cu and leakage conductance G„, as shown on Fig. 1-A, where the letter 

 u indicates the values per unit length. If i denotes the current and 

 V the voltage at a distance x from one end of the line, the well known 

 differential equations are 





(1) 



If we eliminate v from these two equations we have 



LuC^ ~ + {RuCu + aX„) ^ + RuGJ = 1^' • ■ (2) 



Similarly, if i is eliminated, the resulting equation is the same as (2) 

 with i replaced by v. Since we are dealing with simple harmonic 

 forces, the current i varies as cos (co/ + 6) where co is lir times the 

 frequency of vibration and is an arbitrary phase angle. It is 

 usually more convenient to consider i as varying according to the 

 time factor 



i =. |e>C"^+9) = i[cos (co^ + e) +j sin (co/ + 0)], 



where i is the maximum amplitude of i. The solution obtained on 

 this assumption is called the symbolic solution, and the real solution 

 is obtained from the symbolic solution by taking the real part. 

 Substituting the symbolic form of i in equation (2), this equation 

 reduces to 



[(»2L.C + MRX\ + G„L.] + R.G.y ^~ (3) 



The solution for a line can be specified in terms of two parameters, 

 the characteristic impedance and the propagation constant of the 

 line. To show this we note that the solution of (3) is 



i = ^e-i^^ + 5el^^ (4) 



where P = []i?„ + JcoLuJ^Gm + jcoC„] and A and B are constants. 

 From the last of equations (1) we have 



d . 



-lAe-'-' - Be'-'J (5) 



Gu + JwC'u Gu + jojCu 



