418 BELL SYSTEM TECHNICAL JOURNAL 



and the potential energy takes the form 



V = \^ (^-ix_i - A ,x,y + \ Soxo' + ^ ^" (^ o-To - ^2x2)2 



Let 



+ 2 ^ (^2X2 - ^4X4)2 + 2 ^ (^4X4)^. (4) 



XPo 



XPo , 



Oi = -jy- , etc. 



Differentiating the above expressions, substituting into (1), writing 

 X = jcox, .r = -i- , and noting that 62 "= 63 — d = 0, we have 



, a^iA-i^\ . , / - a_i^_i^o\ . 



J^ / \ J^ J 



a_i^o^-i . , /. , I a_i^ 0^ + -So + ai^ 0- \ . 

 x-i + ( jwniQ + ro H j^ ) Xq 



.h = 0, 



Xi = 0, 



aiAoA'. 



J^ 



OiAiAo . , / . , , a:^2^ + 03^2^ \ . a^AiAi 

 — -. Xq + JC0W2 + ^2 H ^ ^'2 -. 



a-2 + jwnii + Ti -\ -. Xi = U, 



J^ \ J^ 



If we were to draw the equivalent circuit from these equations we 

 would find that negative stiffnesses are introduced by the different 

 areas through which the air has to flow. In the shunt arms, however, 

 only positive stiffnesses appear. In order to eliminate the negative 

 stiffnesses it is customary to group the shunt stiffness with a negative 

 stiffness and another positive stiffness into a T structure. It is simple 

 to show that this T structure is equivalent to an ideal auto-transformer 

 shunted by a positive stiffness. The turn ratio of the transformer is 

 given by the ratio of two areas. Of course, we may write for the imped- 

 ance looking into the high side of the auto-transformer 



^" = (zf) ^^' 



where 



Z L = impedance in the low side of auto-transformer. 

 An. m = areas where An > A,,,. 



