no BELL SYSTEM TECHNICAL JOURNAL 



6. Series for 7?'^' When Corner Has No Truncation 



The integrals which appear in the approximations for the reflection 

 coefficients are difficult to evaluate in general. This section serves to put 

 on record several expressions which have been obtained for R^'-^ when the 

 corner is not truncated. Corresponding evaluations of R'^^' would be 

 welcome since the work of Section 7 for small angle corners indicates that 

 j^s) _ 7^(2) is of the same order as R^-K However, I have been unable to 

 go much beyond the results shown here. 



As mentioned in the introduction, H. Levine has studied the effect of a 

 corner in a wave guide by representing it as an equivalent pi network having 

 an inductance for the series element and two equal condensers for the shunt 

 elements. Early in 1947 he derived the following expressions (in our 

 notation) for the elements corresponding to a simpe E corner :* 



Ba/Yo = k 



{^•) 



^r-^n-M-^ 



Bt,/Yo = {kiv)-' cot (/3x/2) 



where Fo is the characteristic admittance of the straight guide, iB^ the 

 admittance of one of the two equal shunt condensers, — iBb the admittance 

 of the series inductance, ^(:v) the logarithmic derivative of r(.v + 1), and 

 /37r is the total angle of the simple corner (for no truncation we set /3 = 2a). 



When the reflection coefficient for the corner is computed from the 

 equivalent network for the case /3 -^ it is found to lie between the approxi- 

 mate value Re^ given by (7-3) and the considerably more accurate value 

 R^E^ given by (7-5). All three approximations are of the form Aff- + 0{^^) 

 where A differs from approximation to approximation but is independent of 

 /3, and 0(/3*) denotes correction terms of order |S^ Since Re^ gives the exact 

 value of .4, it may be regarded as the standard when the three approxima- 

 tions are compared. If this comparison be taken as a guide, it suggests 

 that the rather cumbersome expressions (6-2) and (6-5) for R^e'^ given below 

 are not as accurate as the simpler expressions resulting from Levine's work. 

 Dr. Levine has also obtained corresponding results for the general £-corner 

 of Fig. 1. It is hoped that his work will be published soon. 



When the corner is not truncated it is convenient, as mentioned above, 

 to replace 2q: by ^ so that fiir is the total angle of the bend. For no trunca- 

 tion / = and (2-6) becomes 



gii\ 0) 



chv -+- cos 



- 1. (6-1) 



chv — cos 0_ 

 * I am indebted to Dr. Levine for communicating these expressions to me. 



