190 BELL SYSTEM TECHNICAL JOURNAL 



For small values of /3o//, that is, at low frequencies, very nearly |8 = /3o ; 

 that is, the phase velocity is very near to the velocity of light. The field 

 decays slowly away from the circuit. The longitudinal electric field is small 

 compared with the transverse electric field. In fact, as the frequency ap- 

 proaches zero, the wave approaches a transverse electromagnetic wave 

 traveling with the speed of light. 



At high frequencies the wave falls ofif rapidly away from the circuit, and 

 the transverse and longitudinal components of electric field are almost equal. 

 The wave travels very slowly. As the wavelength gets so short that the 

 spacing / approaches a half wavelength (/3^ = tt) the simple analysis given 

 is no longer valid. Actually, ^( = tt specifies a cutoff frequency; the circuit 

 behaves as a lowpass filter. 



Figure 4.3 shows two opposed sets of fins such as those of Fig. 4.1. Such 

 a circuit propagates two modes, a transverse mode for which the longi- 

 tudinal electric field is zero at the plane of symmetry and a longitudinal 

 mode for which the transverse electric field is zero at the plane of symmetry. 



At low frequencies, the longitudinal mode corresponds to the wave on a 

 loaded transmission line. The fins increase the capacitance between the con- 

 ducting planes to which they are attached but they do not decrease the 

 inductance. Figure 4.6 shows ^h vs. /?o/^ for several ratios of fin height, //, 

 to half -separation, d. The greater is h/d, the slower is the wave (the larger 

 is /3//3o). 



The longitudinal mode is like a transverse magnetic waveguide mode; it 

 propagates only at frequencies above a cutoff frequency, which increases 

 as h/d is increased. Figure 4.7 shows ^h vs. fioh = {<j}/c)h for several values 

 of h/d. The cutoff, for which ^C — tt, occurs for a value of ^qJi less than ir/l. 

 Thus, we see that the longitudinal mode has a band pass characteristic. The 

 behavior of the longitudinal mode is similar to that of a longitudinal mode of 

 the washer-loaded waveguide shown in Fig. 4.8. The circuit of Fig. 4.8 has 

 been proposed for use in traveling-wave tubes. 



The transverse mode of the circuit of Fig. 4.3 can also exist in a circuit 

 consisting of strips such as those of Fig. 4.1 and an opposed conducting 

 plane, as shown in Fig. 4.5. This circuit is analogous in behavior to the disk- 

 on-rod circuit of Fig. 4.9. The circuit of Fig. 4.5 may be thought of as a 

 loaded parallel strip line. That of Fig. 4.9 may be thought of as a loaded 

 coaxial line. 



Wave-analysis makes it possible to evaluate fairly accurately the trans- 

 mission properties of a few simple structures. However, iterated or repeating 

 structures have certain properties in common: the properties of filter 

 networks. 



For instance, a mode of propagation of the loaded waveguide of Fig. 4.10 

 or of the series of coupled resonators of Fig. 4.11 can be represented ac- 

 curately at a single frequency by the ladder networks of Fig. 4.12. Further, 



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