COUPLED HELICES 145 



coupled helices will require similar treatment, namely, Maxwell's equa- 

 tions solved now with the boundary conditions of two cylindrical heli- 

 cally conducting sheaths. As shown on Fig. 2.3, the inner helix is specified 

 by its radius a and the angle 1^1 made by the direction of conductivity 

 with a plane perpendicular to the axis; and the outer helix by its radius 

 h (not to be confused with the mutual coupling coefficient 5) and its 

 corresponding pitch angle i/'-j . We note here that oppositely wound helices 

 require opposite signs for the angles \f/i and i/'o ; and, further, that helices 

 with equal phase velocities will ha\'e pitch angles of about the same 

 absolute magnitude. 



The method of solving Maxwell's equations subject to the above men- 

 tioned boundary conditions is given in Appendix I. We restrict our- 

 selves here to giving some of the results in graphical form. 



The most universally used parameter in traveling-wave tube design is 

 a combination of parameters: 



/3oa cot \pi 



where (So = 27r/Xo , Xo being the free-space wavelength, a the radius of 

 the inner helix, and xpi the pitch angle of the inner helix. The inner helix 

 is chosen here in preference to the outer helix because, in practice, it will 

 be part of a traveling-wave tube, that is to say, inside the tube envelope. 

 Thus, it is not only less accessible and changeable, but determines the 

 important aspects of a traveling-wave tube, such as gain, power output, 

 and efficiency. 



The theory gives solutions in terms of radial propagation constants 

 which we shall denote jt and yt (bj^ analogy with the transverse and 

 longitudinal modes of the transmission line theory). These propagation 

 constants are related to the axial propagation constants ^t and j3( by 



Of course, in transmission line theory there is no such thing as a radial 

 propagation constant. The propagation constant derived there and de- 

 noted r corresponds here to the axial propagation constant j^. By 

 analogy with (2.4.5) the beat phase constant should be written 



How^ever, in practice ^0 is usually much smaller than j3 and Ave can there- 

 fore write with little error 



iSfc = 7e — li 

 for the beat phase constant. For practical purposes it is convenient to 



