COAXIAL AND BALANCED TRANSMISSION LINES 277 



the ratio of the series inductive reactance to the series resistance, the 

 ratio will be 



(2 = .3026V/(log.^)/(^ + 1), • (4) 



where k = b/a. It will be noted that this is the value of Q measured 

 for a short-circuited conductor for low frequencies. As an example, a 

 conductor 3 inches in diameter with the optimum ratio of ^ = 3.6 

 will have a Q oi 3,200 at 100 megacycles. The value of the character- 

 istic impedance Zq of a coaxial line is 



^=60 1oge^- (5) 



For a balanced transmission line the values of R, L, and C are,^ 

 if D is much larger than a, 



„ 83.2 X 10-^ V7 



K = ohms per centimeter, 



L — 4 loge— X 10~^ henries per centimeter length, (6) 



„ 1.111 X 10-^% , . ^ 



C = Y) farads per centimeter, 



4 log,- 



where D is the spacing between wires, and a the radius of one of the 

 pair of wires. With these values, the expressions for Q and Zo become 



Q = .302 V/a loge- ; Zo = 120 log^- ■ (7) 



a a 



Another combination of some interest is obtained by using the inside 

 conductors of two coaxial conductors adjacent to each other. 

 Such a construction results in a balanced and shielded transmission 

 line. All of the constants are double those given by equation (3), 

 except for the capacitance which is halved. 



III. Filters Employing Transmission Lines as Elements 



One of the first uses of transmission lines as elements in wave filters 

 is described in a patent of one of the writers.^ In this patent are con- 

 sidered the characteristics obtainable by combining sections of trans- 



5 U. S. Patent 1,781,469 issued to W. P. Mason. Application filed June 25, 1927; 

 Patent granted Nov. 11, 1930. Wave filters using transmission lines only are very 

 similar to acoustic wave filters as pointed out in an article, "Acoustic Filters," Bell 

 Laboratories Record, April 1928. Most of the equations and results of a former paper 

 on acoustic filters, B. S. T. J., April 1927, p. 258, are also applicable to transmission 

 line filters. 



