/* Negative feedback, finite gain, bandwidth and stability */
kill(all)$
display2d:false$

/* Exact feedback law and large-loop-gain error */
Af : A/(1+A*beta)$
ideal : 1/beta$
relative_error : ratsimp((ideal-Af)/ideal)$
print("Af =",Af)$
print("fractional gain error =",relative_error)$

/* Single-pole op amp A(s)=A0/(1+s/wp) */
As : A0/(1+s/wp)$
Afs : factor(As/(1+beta*As))$
print("single-pole closed-loop transfer =",Afs)$
print("dc gain =",ratsimp(subst(s=0,Afs)))$
print("closed-loop pole =",-(1+A0*beta)*wp)$

/* Numerical example */
A0n : 2.0e5$
ft : 1.0e6$
NG : 100.0$
betan : 1/NG$
Af0 : A0n/(1+A0n*betan)$
fH : (1+A0n*betan)*ft/A0n$
print("finite dc closed-loop gain =",float(Af0))$
print("ideal noise gain =",NG," fractional error =",float(1-Af0/NG))$
print("closed-loop bandwidth (Hz) =",float(fH))$

/* Independent illustrative high-bandwidth two-pole loop used in the figure.
   It is separate from the preceding noise-gain-100 numerical example. */
fp2 : 2.0e5$
loop_db(f) := 100-20*log(f/100)/log(10)-10*log(1+(f/fp2)^2)/log(10)$
fc : find_root(loop_db(f),f,1.0e6,2.0e6)$
phase_deg : -90-atan(fc/fp2)*180/%pi$
phase_margin : 180+phase_deg$
print("independent two-pole loop crossover (Hz) =",float(fc))$
print("loop phase at crossover (degrees) =",float(phase_deg))$
print("phase margin (degrees) =",float(phase_margin))$