8 BELL SYSTEM TECHNICAL JOURNAL 



Similarly, 



--!=4/^ 



By multiplying equation 1.1-1 by 1.1-2 one finds 



-Zil'i = 1 1.1-9 



from which one can compute the exponent px. In the specific circuits 

 shown in Fig. lb 



Zi = Lxpi + Ri and 1.1-10 



Yx - Cxpx 1.1-11 



From 1.1-y, 10 and 11 



?' = - ^' + ^- - - 2X. + -'■ /SFl' '■'-'' 



and by analogy 



p..^-,, + ;.. = - ^1 + ; ^^^ - g 1.1-13 



In equations 1.1-7 and 1.1-8, £io and £20 are ampUtude constants determined 

 by boundary conditions. In equations 1.1-12 and 1.1-13, bx and 62 are the 

 decay or damping constants, coi and C02 the radian frequencies. 

 f^ With finite but loose coupling and small damping the circuits can oscillate 

 with either or both of the two frequencies. 



p^ = p±±i^j^p±^-^r+^^ = P: + 0.5 p,(\ - vrr^^) 1-1-14 

 p, ^ ti^LJt - ^^vnw^ = p-^ + 0.5 px{\ - vrw) 1-1-15 



In the last two equations, the symbol k, defined by '^ = \/ r ^r\, '^' "^^^^^ 



be called the coupling discriminant. The first term of the product on the right 

 side of this expression is the reciprocal of the fractional difference between 

 the uncoupled frequencies; the second term k is the "coupling coefficient." 

 When there is only one coupling impedance, the coupling coefficient is 

 usually defined as the mutual circuit impedance divided by the geometric 

 mean of the separate circuit impedances. A broader definition which 

 applies to all combinations of mutual impedances and admittances is 



k = -%- = -^ 1-1-16 



