HELIX WAVEGUIDE 1375 



constants calculated from the approximate formulas are given to four 

 decimal places, i.e., usually two significant figures. 



The contents of Table I are displayed graphically in Figs. 2(a) through 

 (f), which show plots of (3a vs aa for all modes except TM13 . Repre- 

 sentative values of e are indicated on the curves. Note that the scales 

 are different for the different guide sizes, and that the jSa-scale is com- 

 pressed in all cases. If aa and jSa were plotted on the same scale, the 

 curves would make an initial angle of 45° with the aa-axis when e = 

 constant, or 22.5° when e — e". 



Figs. 3(a) to (f) show the normalized attenuation constants aa of 

 various modes plotted against e" on a log-log scale. In Fig. 3(b) the 

 curves for all TM modes would be similar to the two shown, and in 

 Fig. 3(d) the TM03 curve is like TM12 . Although for some modes the 

 attenuation constant increases steadily as the conductivity decreases 

 over the range of our calculations, in many cases the attenuation passes 

 through a maximum and then decreases as the conductivity is further 

 decreased. This phenomenon will be discussed in Section V. 



It may be noticed that in some instances the limit modes are not 

 unique. For example, Tables 1(a), with e' = 4, and 1(c), with e' = e", 

 for the large guide have in common the case e' = 4, e" = 4. For this 

 case consider the circular magnetic mode corresponding to fia = 

 3.905 -1- 0.344t. If e' is constant (=4) while e" tends to infinity, this 

 mode approaches the TM02 mode in a perfectly conducting guide; but 

 if e' and e tend to infinity while remaining equal to each other, the same 

 mode approaches TMoi in a perfectly conducting guide. Presumably the 

 TMoi-limit mode in the former case coincides with the TMo2-limit mode 

 in the latter case ; but the value of f la for this mode is outside the range 

 of our calculations at e' = e = 4. A similar interchange occurs between 

 the TMii-limit and TMi2-limit modes in the large guide, depending on 

 whether e' is constant or e' tends to infinity with e". There is no evidence 

 of any such phenomenon in the smaller guide of Tables 1(d) and 1(e); 

 but the fact that it can occur means that the limit-mode designations of 

 modes in a lossy waveguide are not entirely unambiguous. The phen- 

 omenon is not due to the presence of the helix, since a helix of zero pitch 

 has no effect on circular magnetic modes. 



Finally it is of interest to compare the propagation constants given 

 by the approximate formula with those obtained by numerical solution 

 of the characteristic equation. A reasonably typical case is provided by 

 the TMo2-limit mode in a 2-inch guide at Xo = 5.4 mm with e = 4, as 

 in Table 1(a). Exact and approximate results for ^a vs aa and aa vs e" 

 are plotted in Fig. 4. As the conductivity decreases, the attenuation con- 



