CURVED WAVE GUIDES 7 



attainable are tolerant to random angular deviations of the order of 

 1 degree. 

 12. For any expected distribution of random angular deviations there 

 exists an optimum wave guide radius for each signal wave length and 

 an optimum signal wave length for each wa\'e guide radius, which 

 minimize the average attenuation. 



References 



1. Jahnke & Emde, Tables of Functions, Dover Publications, New York, 1943. 



2. M. Jouguet, Propagation dans les tujaux courbes, Comptes Rendus — Academie des 



Sciences, Paris, Feb. 18, 1946, March 4, 1946 and Jan. 6, 1947. 



3. M. Jouguet, Effets de la courbure dans un guide a section circulaire, Cables & Trans- 



mission, 1 No. 2, July 1947, pp. 133-153. 



4. S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Company, Inc., New 



York, 1943. 



5. M. Wien, Ueber die Rueckwirkung eines resonierenden Systems, Ann. d. Physik, 



1897, Vol. 61, pp. 151-189. 



ANALYSIS 



1. Interaction of Coupled Circuits 



1.1 Free Oscillations of Coupled Resonators {Fig. IB) 

 The circuits are coupled according to the following four equations: 



ci = —Z\i\ -\- Zmii 1.1-1 



n = YxCx + F„g2 1.1-2 



ei = —Ziii -\r Zrnii 1.1-3 



/o = y^eo + F„ei 1.1-4 



where index i refers to circuit 1, index o to circuit 2 and index ^ to the mutual 

 coupling impedance and admittance. The coupled oscillations have the 

 solution: 



ei = Eue'"'' 4- Elbe"'' 1.1-5 



f2 = E.ae""' + Eobe"" 1.1-6 



In the limiting case of zero coupling (I'm = 0,Zm = 0) the obvious solution 

 shows independent oscillations in the two separate circuits: 



eio = A'l/io = Eioe''^ 1.1-7 



^20 — K^iio = £20 f 1.1-8 



The wave impedance A'l of the primary circuit is found by dividing equation 

 1.1-1 by 1.1-2 



