XOmL\L MODE BEXDS FOR CIRCUK\R ELECTRIC WAVES 1301 



is bent around a fixed support in the center by forces acting on both 

 ends. In order not to exceed elastic deformation the bending radius must 

 not be smaller than 



E 



Rmin = 7 <^1 > (-5) 



Jm&x 



¥ 



where /,aax = flexural stress at elastic limit, 

 E = modulus of elasticity, 

 Oi = outside radius of pipe. 



This minimum bending radius requires a minimum length to change the 

 pipe direction by a specified angle ^o given by 



/.nin = 2eoi?min • (26) 



The total bend loss (24) has been evaluated for a bend configuration as 

 specified by (25) and (26). The result is shoAAii in Fig. 2. The total addi- 

 tional bend loss is only of the order of the TEoi loss in the plain straight 

 ^va^'eguide. For small bending angles the curvature taper becomes .shorter 

 and consequently the mode conversion loss mcreases. The mode conver- 

 sion loss, however, does not go to infinity for zero bending angle. In 

 this ca.se (14) is no longer satisfied, and the mode conversion loss is no 

 longer described by (20). 



The level of the various miwanted modes which can be calculated from 

 (20) is plotted in Fig. 2. 



For a practical waveguide one would decide on a .standard length of 

 dielectric-coated pipe, one or several of which would be inserted whene\'er 

 a change in direction has to be made. Take, in our example, a standard 

 length of 15 feet. With one such section a change of direction up to 15° 

 could be made. For a change in direction up to 30° two such sections 

 would have to be inserted and bent around a fixed support at the center 

 joint. The total loss of Fig. 2 is then a maximum value, which would 

 only occur when the pipe is bent to the highest allowable bending angle. 



IV. A XORiL\L MODE BEXD OF OPTIMUM DESIGX 



The various terms of the total bend loss (24) depend on the bend geom- 

 etry in quite different ways. It is therefore hkely that for a given bending 

 angle ^o a bend geometry can be found, which minimizes the total bend 

 lo.ss. The total bend loss can generallv be written as: 



&"■ 



-4. = SI -{- B y— r:, -1- C J-, ^ , 



/(I — u)- r(w — u-)- 



