Nonlinear fits are difficult and the trick is that you have to provide a reasonable initial guess.
Here is a version of your code which does two fits, one with an approximate initial guess and one with the default initial guess:
import pylab as plb
import matplotlib.pyplot as plt
import matplotlib.axes as ax
import scipy as sp
from scipy.optimize import curve_fit
from matplotlib import rc
import numpy as np
rc('font', **{'family':'sans-serif', 'sans-serif':['Helvetica']})
rc('text', usetex=True)
# Fake data
x = np.arange(0, 70., 2.)
yl = 300 + 63*np.exp(-x/35.)
def func(x, a, b, c):
return a*np.exp(-b*x) + c
popt, pcov = curve_fit(func, x, yl, p0=(40, 0.012, 250), maxfev=20000)
a, b, c = popt
print 'a=', a, 'b=', b, 'c=', c
print 'func=', func(x, a, b, c)
popt2, pcov2 = curve_fit(func, x, yl, p0=None, maxfev=20000)
a2, b2, c2 = popt2
print 'a2=', a2, 'b2=', b2, 'c2=', c2
print 'func=', func(x, a2, b2, c2)
xf = np.linspace(0, 70, 100)
yf = a*np.exp(-b*x) + c
plt.clf()
plt.plot(x, yf, 'r-', label="Fitted Curve")
plt.plot(x, func(x, *popt))
plt.plot(x, func(x, *popt2), 'b-', label='Fit w/o guess')
plt.plot(x, yl, 'go', label='Lacquered')
plt.legend()
plt.ylabel("Temperature (K)")
plt.xlabel("Time (min)")
plt.show()
And here are the resulting fits:
As you can see, the fit with a reasonable initial guess does very well (red line). If you don't provide an initial guess, scipy assumes 1 for all parameters and that works poorly (blue line).