276 BELL SYSTEM TECHNICAL JOURNAL 



very selective networks can be obtained at the high radio frequencies. 

 The effect of the distributed nature of the elements is considered and 

 methods are described for obtaining single-band filters and transform- 

 ers. Experimental measurements of such filters and transformers are 

 shown, and these results indicate that such structures should be of use 

 in short-wave radio circuits. 



II. Characteristics of Transmission Lines 



To facilitate an understanding of the following discussion, the equa- 

 tions of transmission lines as they apply to filter structures will be 

 briefly reviewed first. The equations of propagation for any uniform 

 transmission line can be expressed in the form of equations between the 

 output voltage e^, the output current i^, the input voltage ei, and the 

 input current ii by the relations 



^2 = «i cosh PI — iiZo sinh PI, 



ii = ii cosh PI — -;=- sinh PI, (1) 



where / is the length of the line, P the propagation constant, and Zo 

 the characteristic impedance of the line. In terms of the distributed 

 resistance R per unit length of line, L the distributed inductance, G the 

 distributed conductance, and C the distributed capacitance, P and Zo 

 can be expressed by the relations 



P = ^(R-\-jc.L)(G-j-jo:C); Zo = ^^^j^, (2) 



where co is lir times the frequency. 



The distributed conductance G is usually very low and can be 

 neglected for coaxial or balanced transmission lines in dry atmospheres. 

 For copper coaxial lines, the values of R, L, and Chave been calculated* 

 to be 



R = 41.6 X 10~^ ^ff i — |- T ) ohms per centimeter, 



L — 2 loge- X 1 0~^ henries per centimeter, (3) 



a 



^ 1.11 X 10-^% , ,. , • 



C = r — farads per centimeter, 



2 1oge^ 



where b is the inside radius of the outer conductor and a the outside 

 radius of the inner conductor. If we define the Q of the conductor as 



^ See reference 1, page 415 and page 417. 



