CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 239 



modes is the actual field produced by the actual impressed current. The 

 field is so expressed in (6.44) where the current components /„ are defined 

 by (6.36). 



If it is assumed that there is only one mode of propagation, and if it is 

 assumed that the field is constant over the electron flow, (6.44) can be put 

 in the form shown in (6.47). For waves with a phase velocity small compared 

 with the velocity of light, this reduces to (6.4), which was based on the simple 

 circuit of Fig. 2.3. 



Of course, actual circuits have, besides the one desired active mode, an 

 infinity of passive modes and perhaps other active modes as well. In Chapter 

 VII a way of taking these into account will be pointed out. 



Actual circuits are certainly not lossless, and the fields of the helix, for 

 instance, are not purely transverse magnetic fields. In such a case it is per- 

 haps simplest to assume that the modes of propagation exist and to cal- 

 culate the amount of excitation by energy transfer considerations. This has 

 been done earlier^, at first subject to the error of omitting a term which 

 later- was added. In (6.55) of this chapter, (6.44) is reexpressed in a form 

 suitable for comparison with this earher work, and is found to agree. 



Many circuits are not smooth in the z direction. The writer believes that 

 usually small error will result from ignoring this fact, at least at low signal 

 levels. 



6.1 Excitation of Transverse Magnetic Modes of Propagation by 

 A Longitudinal Current 



We will consider here a system in which the natural modes of propagation 

 are transverse magnetic waves. The circuit of Fig. 4.3, in which a slow wave 

 is produced by finned structures, is an example. We will remember that the 

 modes of propagation derived in Section 4.1 of Chapter IV were of this 

 type. We will consider here that any structure the circuit may have (fins, 

 for instance) is fine enough so that the circuit may be regarded as smooth 

 in the z direction. 



Any transverse electric modes which may exist in the structure will not 

 be excited by longitudinal currents, and hence may be disregarded. 



The analysis presented here will follow Chapter X of Schelkunoff's 

 Electromagnetic Waves. 



The divergence of the magnetic field H is zero. As there is no z component 

 of field, we have 



'J. R. Pierce, "Theory of the Beam-Type Traveling-Wave Tube," Rroc. I.R.E. Vol. 

 35, pp. 111-123, February, 1947. 



^ J. R. Pierce, "Effect of Passive Modes in Traveling-Wave Tubes," Froc. I.R.E., 

 Vol. 36, pp. 993-997, August, 1948. 



