82 BEIJ. SVSl'J-M TECHNICAL JOURN.IL 



cause also of the fact that the mathematical analysis is simplified 

 therei)y. Furthermore any other termination can be regarded and 

 dealt with as an additional terminal impedance, so there is no essen- 

 tial loss of generality invoK^ed. 



A study of the properties of the artificial line is of practical im- 

 portance for several reasons : 



1. The artificial line is often used as a model of an actual trans- 

 mission line and it is therefore of importance to determine theoretically 

 the degree of correspondence between the two. 



2. The solution for the corresponding transmission line with con- 

 tinuously distributed constants is derivable from the solution for the 

 artificial line by keeping the total inductance, resistance, capacity 

 and leakage constant or finite, and letting the number of sections 

 approach infinity. 



3. The artificial line is very closely related, in its properties and 

 performance, to the periodically loaded line, and its solution is, to 

 a first approximation, a working solution for the loaded line. 



4. The structure is of great importance in its own right, and when 

 the impedance elements are properly chosen, constitutes a "wave 

 filter." 



We shall now derive the operational and symbolic equations which 

 formulate the propagation phenomena in the artificial line. Let /„ 

 denote the mesh current in the nth section of the line; /«_i the mesh 

 current in the (w — 1)"' section, etc. Now write down the expression 

 for the voltage drop in the «"' section; in accordance with Kirch- 

 hoff' s law we get : 



(si + 2c.,)/„-So(/„-i + /„+,) =0 (218) 



where, of course, the impedances have the usual significance. 



Now this is a difference equation, as distinguished from a differ- 

 ential equation, but the method of solution is essentialK' the same. 

 We assume a solution of the form 



/„=/le-"i>5e"r (219) 



where A, B and F are independent of n, and substitute in (218). 

 After some simple rearrangements we get 



\ (si + 2zo)-2s2 cosh r(- • Ue-"i'+5e"r; =0. (220) 



Equation (218) is clearly satisfied by the assumed form of solution, 

 and furthermore leaves the constants A and B arbitrary and at our 



