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BELL SYSTEM TECHNICAL JOURNAL 



4.29, 4.30, 4.31 and 4.32 M is given by (4.69), (4.71), (4.72) and (4.73), 

 respectively. 



The factor M may be indifferently regarded as a factor by which we 

 multiply the a-c beam current to give the induced current at the gap, or, 

 as a factor by which we multiply the gap voltage in obtaining the field. We 

 can go further, evaluate E^/^'^P in terms of gap voltage, and use M'/o as the 

 effective current, or we can use the current /o and take the effective field in 

 the impedance parameter as 



£2 = M'\V/(T- 



It is sometimes desirable to make use of a spatial harmonic (w 9^ 0) 

 instead of a fundamental, usually to (1) allow a greater resonator spacing 

 (2) to obtain a positive phase velocity when the fundamental has a negative 

 phase velocity (3) to obtain a phase curve for which the phase angle is 

 nearly a constant times frequency; that is, a phase curve for which the group 

 velocity does not change much with frequency and hence can be matched 

 by the electron velocity over a considerable frequency range. Figure 4.33 

 shows how ^ + 27r (the phase shift per section for m = \) can be nearly a 

 constant times w even when 6 is not. 



l-^i ^ 



Fig. 4.1 — A corrugated or finned circuit with filter-like properties. 



4.1 Field Solutions 



An approximate field analysis will be made for two very simple two- 

 dimensional structures. The first of these, which is shown in Fig. 4.1, is 

 empty space for y > 1 and consists of very thin conducting partitions in the 

 y direction from y = to y = — //; the partitions are connected together 

 by a conductor in the z direction at y = — //. These conducting {)artitions 

 are spaced a distance C apart in the z direction. The structure is assumed to 

 extend infinitely in the -\-x and —x directions. 



In our analysis we will initially assume that the wavelength of the propa- 

 gated wave is long compared with (. In this case, the effect of the partitions 

 is to prevent the existence of any y component of electric field below the z 

 axis, and the conductor at y = —h makes the s component of electric held 

 zero at y = —z. 



In some perfectly conducting structures the waves propagated are either 

 transverse electric (no electric lield component in the direction of propaga- 

 tion, that is, z direction) or transverse magnetic (no magnetic field com- 



