1248 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



round guide causes coupling l)etween the circular electric wave (TEoi) 

 and the TMu wave, and in round solid pipe the TEoi and TMu wa^'es 

 are degenerate, i.e., they have identical phase constants. This degen- 

 eracy has the effect of bringing in jihase all components transferred to 

 TMu 110 matter how gradually the bend might take place. Theory 

 neglecting dissipation ' shows that in a round pipe bent with any radius 

 of curvature, the power will flow from th(^ TEoi mode to the TMu mode 

 and back as a function of the total bend angle 6 according to the relations 



TEoi amplitude = cos f ^ - j (9) 



TMii" amplitude = sin (^^ j\ (10) 



where 



TT ^o 



2^0" 



Z.6Z a 



Xo = free-space wavelength 

 a = waveguide raduis 



A solid round pipe is unsatisfactory for transmission of the circular 

 electric wave around bends in the broad bands we seek to use. 



One method of making the guide satisfactory in bends is to break the 

 degeneracy between the TEoi and TMu waves. Use of an elliptical pipe 

 has been shown theoretically to be one way of doing this. For a 2" di- 

 ameter guide at 50,000 mc an eccentricity of about 0.3 permits a bending 

 radius of 700 feet with theoretical bend losses in the range to 0.17 db 

 for any total bend angle; the heat loss coefficient for such an elliptic 

 guide is about 35 per cent higher than for a perfectly round guide.^ 



Another method of avoiding bend losses is to introduce dissipation to 

 the TMii wave without adding loss to the TEoi w^ave. It has been shown 

 that a large difference between the attenuation coefficients of two coupled 

 waves reduces the power transferred from the low-loss wave to the 

 high-loss wave. Applied to the bend problem, this means that a struc- 

 ture with increased TMu loss may be bent with less signal (TEqi) loss 

 even though the phase constants might be degenerate. The reader is 

 referred to the earlier paper for a more complete discussion. There exist 

 several alternate forms of circular electric waveguide (to be discussed) 

 which have an attenuation coefficient for TMn more than 5,000 times 

 the attenuation coefficient for TEoi • The calculated extra loss in the 

 bend region for such structures and for solid round pipe has been plotted 



