/*
   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 non-identical
*/

/*ABCD matrices for the three RC stages*/
ABCD1: matrix([1+1/(R1*C1*s),1/(C1*s)],[1/R1,1])$
ABCD2: matrix([1+1/(R2*C2*s),1/(C2*s)],[1/R2,1])$
ABCD3: matrix([1+1/(R3*C3*s),1/(C3*s)],[1/R3,1])$
assume(R1>0,R2>0,R3>0,C1>0,C2>0,C3>0)$

/*Obtain transfer function of overall RC phase shifter network*/
ABCD: ABCD1.ABCD2.ABCD3$
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);