/* Oscillator principles: symbolic checks and numerical practice */
kill(all)$ display2d:false$ fpprintprec:10$
print("OSCILLATOR PRINCIPLES, LOOP GAIN AND STABILITY")$

/* Positive-feedback characteristic equation. */
H : A/(1-A*beta)$
print("Closed-loop transfer A/(1-A beta) =",H)$
print("Characteristic equation: 1-A beta = 0")$

/* Cubic amplitude equation da/dt=(mu-nu*a^2)*a. */
as : sqrt(mu/nu)$
stability_slope : ratsimp(subst(a=as,diff((mu-nu*a^2)*a,a)))$
print("Steady nonzero amplitude =",as)$
print("Linearized envelope slope at steady amplitude =",stability_slope)$

/* Sensitivity of an LC frequency. */
w0 : 1/sqrt(L*C)$
SL : ratsimp(diff(w0,L)*L/w0)$
SC : ratsimp(diff(w0,C)*C/w0)$
print("Normalized L sensitivity =",SL)$
print("Normalized C sensitivity =",SC)$

/* Numerical example. */
Lnum:10e-3$ Cnum:10e-9$
fnum:float(1/(2*%pi*sqrt(Lnum*Cnum)))$
frac_shift:float(-0.5*(0.005+0.005))$
print("For L=10 mH, C=10 nF, f0 (Hz) =",fnum)$
print("If both rise 0.5 percent, Delta f/f =",frac_shift)$
quit();