/*
   This code is released into the public domain

   This maxima code works out the critical gain and
   oscillation frequency of a three stage RC phase shift
   oscillator where the three RC stages are identical
*/

/*ABCD matrices for the three RC stages*/
ABCD: matrix([1+1/(R*C*s),1/(C*s)],[1/R,1])$
assume(R>0,C>0)$

/*Obtain transfer function of overall RC phase shifter network*/
ABCD: ABCD.ABCD.ABCD$
ABCD: ABCD,s=%i*w$
1/list_matrix_entries(ABCD)[1]$
H: ratsimp(%)$

/*Calculate oscillation frequency*/
solve(imagpart(H),w)$
map(lambda([eq],rhs(eq)),%)$
sublist(%,lambda([x],is (x>0)))$
w0: %[1]$
f0: w0/(2*%pi)$

/*Calculate critical gain*/
H,w=w0$
G: combine(expand(ratsimp(-1/%)))$

/*Print results*/
display(G,f0);