1260 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



manner. Assuming the conductivity to be high though finite and the 

 loss factor of the dielectric material to be small, we take the field pattern 

 and wall currents of the lossless case and calculate the total transmitted 

 power P and the power P m absorbed per unit length of the waveguide by 

 the metal walls of finite conductivity. The wall current attenuation <xm 

 is then given by 



IPm 



(12) 



The result of this calculation as carried through in Appendix II is: 



au _ X2 / x^ — x^ / 1 I !!1 V. 



r. ~'n24/^2_,^2V"^9 ^ 

 Oi-om Vom y X2 — i:X\ \^ L 



r 2 

 X^ 



U\ {Xi , pX-y) -^Ri (X2 , pX^ 



Rl{x2,pX2) > 



(13) 



In this expression «>/ is related to the TE^m attenuation constant aom 

 of the plain waveguide. 



For 1 — p = 5 « 1 we introduce the series expansions of the functions 

 Ui(x2, pXi) and Ri{x2, PX2), 



^ = (e - 1) ^ 8\ (14) 



(Xom Vom 



Here Aa^ is defined as the change in wall current attenuation compared 

 to the attenuation in the plain waveguide. 



III. PROPERTIES OF COUPLED TRANSMISSION LINES 



Wave propagation in gentle bends of a round waveguide can be de- 

 scribed in terms of normal modes of the straight guide. ^ The bend causes 

 coupling between the normal modes. The TEoi wave couples to the TMn 

 wave and to the TEi„ waves and the propagation in the bend is de- 

 scribed by an infinite set of simultaneous linear differential equations. 

 An adequate approximate treatment is to consider only coupling between 

 TEoi and one of the spurious modes at a time. Furthermore, only the 

 forward waves need to be considered, since the relative power coupled 

 from the forward waves into the backward waves is quite small. Thus, 

 the infinite set of equations reduces to the well known coupled fine 

 equations: 



^ + 71^1 - JcE2 = 0, 

 dz 



(15) 



^ + 72^2 - JCE, = 0, 



dz 



