20 BELL SYSTEM TECHNICAL JOURNAL 



CHAPTER III 



THE HELIX 



Synopsis of Chapter 



ANY circuit capable of propagating a slow electromagnetic wave can be 

 used in a traveling-wave tube. The circuit most often used is the helix. 

 The helix is easy to construct. In addition, it is a very good circuit. It has a 

 high impedance and a phase velocity that is almost constant over a wide 

 frequency range. 



In this chapter various properties of helices are discussed. An approximate 

 expression for helix properties can be obtained by calculating the properties, 

 not of a helix, but of a heUcally conducting cylindrical sheet of the same 

 radius and pitch as the helix. An analysis of such a sheet is carried out in 

 AppendLx II and the results are discussed in the text. 



Parameters which enter into the expressions are the free-space phase con- 

 stant |So = w/c, the axial phase constant /3 = w/v, where v is the phase 

 velocity of the wave, and the radial phase constant 7. The arguments of 

 various Bessel functions are, for instance, yr and 7c, where r is the radial 

 coordinate and a is radius of the helix. The parameters /So, jS and 7 are 

 related by 



/32 = /35 -f 7' 



For tightly wound helices in which the phase velocity v is small compared 

 with the velocity of light, 7 is very nearly equal to jS. For instance, at a 

 velocity corresponding to that of 1,000 volt electrons, 7 and /? differ by 

 only 0.4%. 



Figure 3.1 illustrates two parameters of the helically conducting sheet, 

 the radius a and pitch angle ^l/. For an actual helix, a will be taken to mean 

 the mean radius, the radius to the center of the wire. 



Figure 3.2 shows a single curve which enables one to obtain 7, and hence 

 /3, for any value of the parameter 



coa cot i/' 



Po a cot \p = . 



c 



This parameter is proportional to frequency. The curve is an approximate 

 representation of velocity vs. frequency. At high frequencies 7 approaches 



