504 



BELL SYSTEM TECHNICAL JOURNAL 



The corresponding potential of the wire is 



V = Ke~^' -^ + 2Z /' "^^ ■'^^^^^'" 



+ Xe^^ 



A 



^2kX 



dyf(y)e-y^ 



(27) 



+ F(x). 



In these equations, y and K are the propagation constant and the c^ar- 

 acteristic impedance of the circuit ^ consisting of the wire with ground 

 return, while A and A' are arbitrary or integration constants which 

 are determined from the boundary conditions. It will be observed 

 that if the arbitrary impressed field is removed (/ = -F = 0) , the solution 

 reduces to the usual form. If the terminal impedances are specified, 

 it follows from (26) and (27) that the problem is completely solvable 

 provided that/ is specified along the wire and FaX its physical terminals. 

 Two more general cases of practical importance will next be formu- 

 lated : 



(1) Balanced Pair of Wires 



Let Ki, 7i be the characteristic impedance and propagation constant 

 of transmission over the metallic circuit; and Kz, 73 the corresponding 

 quantities for the case of the two wires in multiple, with ground return. 

 Let /i and /2 be the electric force of the primary or impressed field 

 along the surfaces of the wires No. 1 and No. 2 respectively, and /i 

 and li the currents in the wires. The solution may then be written as 



where 



- eT'^" 



(28) 



Ix = a -\- c, I2 = — a -\- c, 



^ + 2^ J' {My) - My) } e^^Hy ] 



^' + 2k/' ^^'^^"^ - My)]e-'^'dy]^ . 

 c = e-T3- 1^ C + ^ J% {/i(3') + My) ] e'^'dy j 



- .-3X j^ c' + ^ J% [My) + My) \ e-'^'dy ] . 



The component a corresponds to transmission over the metallic or 

 physical circuit; while the component c corresponds to transmission 

 over the two wires in multiple, with ground return. A and C are the 



8 It will be observed that in these equations the characteristics of the ground do 

 not appear explicitly. They are, however, implicitly involved in K and 7 of the 

 ground return circuit. 



