TRAVELING-WAVE TUBES 33 



to the direction of conduction and flattening it out. This gives us the plane 

 conducting sheet shown in Fig. 3.8. The indicated coordinates are z to the 

 right and y upward: x is positive into the paper. The fields about the de- 

 veloped sheet approximate those about the helically conducting sheet for 

 distances always small compared with the original radius of curvature. 



The straight dashed line shown on the helix impedance curve of Fig. 3.7 

 can be obtained as a solution for the "developed helix." We see that it is 

 within 10% of the true curve for values of ya greater than 2.8. We might note 

 that a 10% error in impedance means only a 3^% error in the gain 

 parameter C. 



In solving for the fields around the sheet, the developed surface can be 

 extended indefinitely in the plus and minus y directions. In order that the 

 fields may match when the sheet is rolled up, they must be the same at 

 y = 0, 3 = lira sin \J/ and y = lira cos \p, z — Q. The appropriate solutions 

 are plane electromagnetic waves traveling in the y direction with the speed 

 of light. 



For positive values of .v, the appropriate electric and magnetic fields are 



£. = jE,e~''' e-''' e''^'' (3.14) 



£, = 



We should note that the x and s components of the field can be obtained 

 as gradients of a function 



^ = -{E,h)e-'' e-^'' e-^^'" (3.15) 



I where 



-Ex = -d^/dz 



(3.16) 

 £. = -d^/dy 



d'^^/dx' + d'-^/dz'' = (3.17) 



i 

 Thus, in the xz plane, $ satisfies Laplace's equation. 



The magnetic field is given by the curl of the electric field times j'/w/x. 



Its components are: 



IXC 



H, = — Eoe-'^'e-'^'e''^'" (3.18) 



uc 



^ Maxwell's equations are given in Appendix I. 



