164 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



For the sake of definiteness, let us choose actual figures: let /3a = 2.0. 

 and hi a = 1.5. And let us, arbitrarily, demand that R always be less than 

 -20 db. 



This gives sin (7r5/2) < 0.316 and 7r5/2 < 18.42° or 0.294 radians, 

 8 < 0.205. With the optimum value of (Sea = 1.47, this gives the mini- 

 mum permissible value of I3ca of 1.47/(1 + 0.205) = 1.22. From the 

 graph on Fig. 2.2 this corresponds to values of jSa of 1.00 and 3.50. 

 Therefore, the reflected power is down 20 db over a frequency range of 

 aj2/aji = 3,5 to one. Over the same range, the directivity is better than 

 10 to one. Suppose a directivity of better than 20 db were required. 

 This requires sin (7r5/2) = 0.10, 8 = 0.0638 and is obtained over a fre- 

 quency range of approximately two to one. Over the same range, the 

 reflected power would be down by 40 db. 



In the above example the full bandwidth possibilities have not been 

 used since the coupler has been assumed to have optimum length when 

 jSctt is maximum. If the coupler is made longer so that when I3ca is maxi- 

 mum it is electrically short of optimum to the extent permissible by 

 the quality requirements, then the minimum allowable (S^a becomes even 

 smaller. Thus, for h/a =1.5 and directivity 20 db or greater the rea- 

 lizable bandwidth is nearly three to one. 



When the coupling helix is non-reflectively terminated at both ends, 

 either by means of two coaxial lines or a coaxial line at one end and a 

 resistive element at the other, the directivity is, ideally, infinite, irrespec- 

 tive of frequency; and, similarly, there will be no reflections. The power 

 transfer to the inner helix is simply proportional to cos (t8/2). Thus, 

 under the conditions chosen for the example given above, the coupled- 

 helix transducer can approach the ideal transducer over a considerable 

 range of frequencies. 



So far, we have inspected the performance and bandwith of the 

 coupled-helix transducer from the most optimistic theoretical point of 

 view. Although a more realistic approach does not change the essence 

 of our conclusions, it does modify them. For instance, we have neglected 

 dispersion on the helices. Dispersion tends to reduce the maximum at- 

 tainable bandwidth as can be seen if Fig. 2.4.2 rather than Fig. 2.2 is 

 used in the example cited above. The dielectric that exists in the annular 

 region between coupled concentric helices in most practical couplers 

 may also affect the bandwidth. 



In practice, the performance^ of coupled-hc^lix transducers has been 

 short of the ideal. In the first place, the match from a coaxial line to a 

 helix is not perfect. Secondly, a not inappreciable fraction of the RF 

 power on a real wire helix is propagated in the form of spatial harmonic 



