1150 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



that is, when the thickness of the conducting layers is twice the thickness 

 of the insulating layers. Thus the result obtained in Section IV for extreme 

 Clogston 1 lines is shown to hold for Clogston lines with an arbitrary 

 degree of filling, provided only that the permeabilities of the conducting 

 and insulating layers are equal. 



We emphasize that the preceding results apply only when the layers 

 are infinitesimally thin. If the layers are of finite thickness, then the 

 optimum value of 6 will be less than that calculated for infinitesimally 

 thin layers. The case of finite layers will be discussed in Section XI. 



X. HIGHER MODES IN CLOGSTON LINES 



We shall now investigate certain of the higher modes which are possible 

 in Clogston-type transmission lines. As elsewhere in this paper, we shall 

 restrict ourselves to modes having H^ , Ey , Ez or H^ , Ep , Ez field 

 components only, and for simplicity we shall assume stacks of infinitesi- 

 mally thin laminae backed by high -impedance boundaries; but we shall 

 place no restrictions on the relative thicknesses of the stacks and the 

 main dielectric. We shall suppose, however, that the main dielectric 

 always satisfies Clogston's condition. From physical considerations we 

 anticipate the existence of higher modes of two types: 



(1) In a partially filled Clogston line containing a finite thickness of 

 main dielectric, there will be a group of modes very similar to the modes 

 which can propagate between perfect conductors when the frequency is 

 high enough to allow one or more field reversals in the space between 

 the conductors. In a Clogston line these modes will have most of their 

 field energy in the main dielectric, and for lack of a better term may be 

 called "dielectric modes". They will all be cut off at sufficiently low 

 frequencies, and for this reason are not likely to be of much engineering 

 importance. The cutoff frequency of any particular dielectric mode is 

 approximately inversely proportional to the thickness of the main di- 

 electric, so that these modes cannot exist in a completely filled Clogston 2. 



(2) There will also be a group of modes which are closely bound up 

 with the laminated stacks, and which correspond to one or more current 

 reversals in the stacks themselves; we shall call these the "stack 

 modes".^^ The stack modes will propagate do^^^l to zero frequency on 

 either a partially or a completely filled Clogston line. They mil have 

 higher attenuation constants than the principal mode, but occasions 

 may arise in which they are of considerable practical importance. We 

 shall therefore consider these modes in some detail in what follows. 



2^ The stack modes in plane lines were discussed bv Clogston in Reference 1, 

 Sections IV-VI. 



