32 



BELL SYSTEM TECHNICAL JOURNAL 



and in the whole range shown the impedance differs from this value by a 

 factor less than 1.5. 



We might have defined a "longitudinal" voltage Vi as half of the integral 

 of the longitudinal component of electric field at the surface of the helically 

 conducting sheet for a half wavelength (between successive points of zero 

 field). We find that 



Vt = Vl - {v/cY V, = (7//3)F, 



and, accordingly, the "longitudinal impedance" K( will be 

 K( = [1 - {v/cy]K, = (ym'Kt 



(3.12) 



(3.13; 



Our impedance parameter, E~/0~P, is just twice this "longitudinal 

 impedance." 



^CIRCUMFERENTIAL 

 ''; CIRCLES 



b \ b b 



DIRECTION 

 OF AXIS ^., 



-2 7ra SIN t/^ 



Fig. 3.8 — A "developed" helically conducting sheet. The sheet has been slit along a 

 line normal to the direction of conduction and flattened out. 



The transverse voltage Vi is greater than the longitudinal voltage Vf 

 because of the circumferential magnetic flu.x outside of the heli.x. For slow 

 waves V.C is nearly equal to Vt and the fields are nearly curl-free solutions of 

 Laplace's equation. In this case the circumferential magnetic flux is small 

 compared with the longitudinal flux inside of the helix. 



For the circuit of Fig. 2.3 the transverse and longitudinal voltages are 

 equal, and it is interesting to note that this is approximately true for slow 

 waves on a helix. For very fast waves, the longitudinal voltage becomes small 

 compared with the transverse voltage. 



For a typical 4,000-megacycle tube, for which ya = 2.8, Fig. 5 indicates a 

 value of Ki of about 150 ohms. 



3.2 The Developed Helix 



For large helices, i.e., for large values of ya, the fields fall off very rapidly 

 away from the wire. Under these circumstances we can obtain quite accurate 

 results by slitting the helically conducting sheet along a spiral line normal 



