1356 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



dix. In practice, the range of validity of these expressions is usually 

 limited by the criterion 



/1 2\l/2 



^^^"^^ \a + iA^\«l (8) 



V 



The numerical calculations described in Section IV indicate that the 

 approximations are good so long as the left-hand side of (8) is less than 

 about 0.1, and that they break down a little sooner for TE modes than 

 for TM modes. 



Inspection of (7) reveals three cases of particular interest, namely 

 ^ = 0°,\p = tan~^ w(l — v^f^/pp, and i/- = 90°. These cases, which were 

 mentioned in Section II and are discussed again below, correspond to 

 preferential propagation of certain modes, in which the wall currents 

 follow the direction of the conducting helix. The preferred modes have 

 zero attenuation in the present treatment because the helical sheath is 

 assumed to be perfectly conducting. In practical helices wound from 

 insulated copper wire the loss should be only slightly greater than in 

 round copper pipe of the same diameter. The slight increase (of magni- 

 tude 10 per cent to 30 per cent) is due to the slightly nonuniform cur- 

 rent distribution in the wires, an effect that can be kept small by keeping 

 the gaps between the wires of the helix small. In general the attenuation 

 constants of modes whose wall currents do not follow the helix are orders 

 of magnitude larger than the attenuation constants of the preferred 

 modes. 



iA = 0° 



The circular electric (TEom) modes have attenuation constants sub- 

 stantially the same as in solid copper pipe. The additional TEom loss if 

 the pitch angle is not quite zero is proportional to tan^ \{/. This added loss 

 can be made very small by using fine wire for winding the helix. 



The losses for the unwanted modes can be made large by a proper 

 choice of jacket material. When ^ = 0, equations (7) yield 



TM„„j modes 



a(l — v^y^ 



TE„m modes 



a + iA/3 = i^ 'J- -^^ (^ + iv) (9b) 



a p^ — n^ 



