TRAVELING-WAVE TUBES 35 



The power flow along the axis is that crossing a circumferential circle, 

 represented by lines a-b in Fig. 3.8. As the power flows in the y direction, 

 this is the power associated with a distance lira sin \[/ in z direction. Also, 

 the power flow in the +.v region will be equal to the power flow in the —x 

 region. Hence, the power flow in the helix will be twice that in the region 

 X = to .V = + 20 , c = to s = lira sin i/'. 



P = 2 / / (l)(E,H* - E,Ht) dx d% (3.26) 



Jz=0 •'2=0 



This is easily integrated to give 



P = 27ra sin ^pE, ^3 27) 



The magnitude E of the axial component of field is 



£ = £o cos yp (3.28) 



Using (3.21), (3.22), (3.24) and (3.28) in connection with (3.27) we obtain 



(£y/3'^P) = (7//3)H/3/i3o)(M^/27r7a) (3.29) 



We have 



Thus 



nc = m/a/wc = vW^ = 377 ohms 



£2//32p = (7//3)''(^/^o)(60/7a) (3.30) 



The longitudinal impedance is half this, and the transverse impedance is 

 (8/7)- times the longitudinal impedance. 



Z.?) Effect of Wire Size 



An actual hehx of round wire, as used in traveling-wave tubes, will of 

 course differ somewhat in properties from the helically conducting sheet 

 for which the foregoing material applies. 



One might expect a small difference if there were many turns per wave- 

 length, but actual tubes often have only a few turns per wavelength. For 

 instance, a typical 4,000 mc tube has about 4.8 turns per wavelength, while 

 a tube designed for 6 mm operation has 2.4 turns per wavelength. 



If the wire is made very small there will be much electric and magnetic 

 energy very close to the wire, which is not associated with the desired field 

 component (that which varies as exp(— jjSs) in the z direction). If the wire 

 is very large the internal diameter of the helix becomes considerably less 

 than the mean diameter, and the space available for electron flow is reduced. 

 As the field for the helically conducting sheet is greatest at the sheet, this 



