4 BELL SYSTEM TECHNICAL JOURNAL 



consisting of single capacity C, and an impedance Z which includes an 

 inductance L and a damping resistance R. The resonators are coupled 

 by a small mutual inductance Z„, and by a small mutual capacity Ym ■ 



The behavior of coupled resonators is very well known to radio engineers. 

 They occur as tuned transformers in amplifier circuits, as band-pass filters 

 and as "tank circuits" in radio transmitters. Even before the advent of 

 radio, their acoustical equivalents were studied in the form of resonant 

 tuning forks. The mathematical aspects of this problem were already 

 clearly set forth in a paper by \Men written in 1897^ He showed that the 

 interaction between the free vibrations of two tuned circuits depends on 

 the coupling coefficient and on the ratio of their complex resonance fre- 

 quencies. The closer the two frequencies are to each other, the less coupling 

 is needed to transfer energy between the two circuits. The reason is that 

 the individual free vibrations of two nearly synchronous circuits remain in 

 step long enough to accumulate the small energy transfer impulses of many 

 vibrations. 



Now consider the two transmission lines of Fig. 2a and assume that a 

 constant frequency signal is impressed upon the input of one or both of them. 

 The signals are carried along the two lines as traveling waves. Again it is 

 true that loosely coupled signals affect each other strongly if they remain in 

 step. With traveling waves "remaining in step" means that they must 

 travel with approximately equal phase velocities. We conclude that the 

 phase velocities or phase constants of coupled transmission lines play a 

 similar role as the resonant frequencies of coupled tuned circuits. This 

 intuitive reasoning is confirmed by analysis (see Section 1 of the analytical 

 part of this paper). 



We thus find that we must expect trouble for TEoi wave guide trans- 

 mission if a mode with an azimuth index 1 has a propagation constant 

 close to that of the TEoi . It so happens that there exists one mode, the 

 TMu , which in an ideal wave guide has exactly the same propagation 

 constant as the TEoi . This then should be the principal source of trouble — 

 and from previous work it is known that such is the case. 



Our discussion of coupled transmission lines has shown that the interaction 

 effects are functions of their relative uncoupled propagation constants and 

 of the coupling coeflicient. The propagation constants of the TEoi and 

 TMu wave guide modes are known but their coupling coefficient remains 

 to be found. 



Since the energy of the transmission modes is located in the dielectric 

 inside the wave guide, we consider first the coupling between the plane 

 "slant wave" groups from which the modes are built up. 



^ Reference 5. 



