462 BELL SYSTEM TECHNICAL JOURNAL 



are significant parameters of the system and, for the present, need 

 no further expansion. 



Another general formula for ju^S is obtained by making use of the 

 image parameters of the coupling network. If the image impedances 

 at terminals 1, 2, and terminals 3, 4, are denoted by Ki and K^, 

 respectively, and the image transfer constant by d, then 



^^ ~ {K^K^ + R1R2) sinh e + {RiKi + R^K^) cosh d ' ^^ 



The two sets of parameters are related by the equations 



All All. 



22 



^22 '-in, 22 



tanh^0 = ^^^ 

 A11A22 



Equation (6) is useful in many cases because of the fact that the fre- 

 quency characteristics of the image parameters are well known for a 

 large number of circuit configurations, particularly those of wave 

 filters. 



In dealing with many practical oscillator circuits, the simplifying 

 assumption may be made that the coupling network contains only pure 

 reactances. The determinants in equation (5) then become either real 

 quantities or pure imaginaries, thus making it easy to separate the real 

 and the imaginary parts of ^u/?. If the number of meshes in the jS cir- 

 cuit, or the number of rows or columns in the determinant A, is even, 

 then A will be real and if the number is odd A will be imaginary. The 

 determinant An, 22 will be of the same character as A, but will take 

 the opposite sign, and determinants An, A22, and A21 will be imaginary 

 when A is real and real when A is imaginary. Accordingly, equation 

 (5) may be transformed to 



^^ {R^Du + R2D,,) + j{D - RrR^Du, 22) ' ^ ^ 



in which the D's are determinants of the mesh reactances correspond- 

 ing respectively to the A's of equation (5) having the same subscripts. 

 The phase angle of ^t/3, denoted by tp, is given by 



D — RiRiDn, 22 /f.\ 



