CIRCULAR ELECTRIC WAVE IN SERPENTINE BENDS 



1287 



A numerical example shows that the increase in TEoi attenuation caused 

 by coupHng to the TMn wave in serpentine bends is small. For a copper 

 pipe with 2.00-inch I.D. and 2|-inch O.D. and a supporting length 

 I = 15 ft, we have d = 1.51 X 10~", and at X = 5.4 mm we get Aas/aoi = 

 0.19 X 10"'. 



For coupling between the TEoi wave and the waves of the TEi„ family 

 the difference in propagation constant is no longer small; the approxima- 

 tion which was valid for the TMn wave can not be made for TEi„ waves. 

 Actually, the supporting distance is usually several beat wavelengths. 

 Therefore, no essential simplifications of (21) are possible for the general 

 case of coupling between TEoi and TEi„ . But closer examination of (21) 

 shows that if 



2A(3l = 2m7r 



m = 1, 2, 3, 



(26) 



is satisfied, the difference Tn-T22 becomes very small. The net power 

 converted to any of the TEi„ modes may be small in each section of the 

 iterative serpentine bends, but if (26) is satisfied the contributions from 

 each section add in phase and in a long line with the square of distance 

 more and more power is built up in the particular TEi„ wave. Only when 

 the attenuation in this TEi„ wave is large enough to damp out the power 

 as fast as it is converted will an undistorted TEoi propagation be main- 

 tained. This condition for the attenuation constant can be derived from 

 I ^11 - 7^22 I '» 4 I 7^12^21 I . If I Aj8 I » I Aa I and I Aal | » Co'(f/10mir 

 then Tn - Tn - -2Aale~^'\ and since ^12^21 = -9 Co'(f/rnT e~^^'^ 

 the condition for undistorted TEoi propagation is: 





(27) 



If (27) is satisfied the wave propagation is again described by (17). 

 Neglecting all terms which are small because of (27) the wave ampli- 

 tudes are: 



Eln = 



1 +9 



2 ,2 

 Cod 



1 



E.n = -.;3 



m V 2Aa/ 

 Cod 1 



—yi'il 



+ 9 



1 



2j2 



Cod 



mV^ 4AaH 



... -C' \^ 



—ixnl 



(28) 



[1 - e Je 



-Ti'ii 



mV 2Aa/ 

 For not too large values of n, the first term of E^ may be written: 



1 + 9 



Co d 



1 



2Aal 



-Jinl 



ini 



exp 



- 71 - 9 



1 



2 j2 



Co a 



WiV 2Aa/2 



nl 



The additional attenuation to the Ei wave as caused by the continuous 



