11. Rétroaction et oscillations
PDF : 
11 retroactions oscillations (6.81 Mo)
PPT : 
Retroaction et oscillations (200.7 Ko)
CODE :
In [1]:
import numpy as np
import numpy.random as npr
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.integrate import odeint
from scipy import signal
import pandas as pd
In [2]:
# Definition de la fonction affine
def f_aff(x, a, b):
return a*np.log10(x)+b
# Montecarlo
def montecarlo(X, Y, uX, uY, N_mc=100, color='r', mini=None, maxi=None):
sigma_X = uX*np.ones(X.size)
sigma_Y = uY*np.ones(Y.size)
param = np.zeros((2,N_mc))
for i in range(N_mc):
x_mc = npr.normal(loc=X, scale=sigma_X)
y_mc = npr.normal(loc=Y, scale=sigma_Y)
pop, pcov = curve_fit(f_aff, x_mc, y_mc)
param[0,i] = pop[0]
param[1,i] = pop[1]
#mini = mini if mini else X.min()
#maxi = maxi if maxi else X.max()
#x_th = np.linspace(mini, maxi, 4)
# y_th = f_aff(x_th, *pop)
# for i in range(N_mc):
# y_th = f_aff(x_th, *param[:,i])
# plt.plot(x_th, y_th, color, alpha=0.05)
#plt.errorbar(X, Y, xerr=uX, yerr=uY, fmt='+', capsize=2, color=color)
# plt.show()
a_moy = np.mean(param[0,:])
b_moy = np.mean(param[1,:])
x = np.linspace(list(X)[0],list(X)[len(X)-1],100)
# plt.plot(x,f_aff(x,a_moy,b_moy),'b--')
# a_moy = np.mean(param[0,:])
# print("La pente moyenne sur les ", N_mc, "mesures est de :", np.round(a_moy,4))
a_sig = np.std(param[0,:]) #ecart type
b_sig = np.std(param[1,:]) #ecart type
# print("L'écart type entre les pentes des", N_mc, "mesures est de :", np.round(a_sig,4))
# print("Le résultat est donc de :", f"a = {a_moy:.3f} +/- {a_sig:.3f}")
# print("Cela correspond à une erreur de:", f"a_rel = {a_sig/a_moy*100:.0f} %")
return (a_moy, a_sig, b_moy, b_sig)
Diagramme de Bode d'un AOP¶
In [3]:
# Données
mu0 = 2e5
tau0 = 3e-2 # en s
# Fonction de transfert
ft = signal.lti([mu0], [tau0, 1])
# Calcul du diagramme de Bode
omega = np.logspace(0.8, 6, 1000)
w, mag, phase = signal.bode(ft, w=omega)
# Affichage
plt.subplot(211)
plt.title('Diagrammes de Bode')
plt.semilogx(w, mag, lw=2) # Amplitude
#plt.xlabel('Pulsation (rad/s)')
plt.ylabel('Amplitude (dB)')
plt.grid(which='both')
plt.subplot(212)
plt.semilogx(w, phase) # Phase
plt.xlabel('Pulsation (rad/s)')
plt.ylabel('Phase (deg)')
plt.grid(which='both')
plt.tight_layout() # Ajuster le placement des courbes
plt.show()
Diagramme de Bode du montage amplificateur non inverseur¶
In [4]:
# Données
R1 = 479 # en ohm
R2 = 1.01e3 # en ohm
H0 = mu0 / (1 + mu0 * R1/(R1 + R2)) # gain de l'AOP
tauBF = H0 * tau0 / mu0
print(H0, tauBF)
In [5]:
# Fonction de transfert
ft = signal.lti([H0], [tauBF, 1])
# Calcul du diagramme de Bode théorique
omega = np.logspace(2, 8, 1000)
w, mag, phase = signal.bode(ft, w=omega)
In [13]:
# Diagramme de Bode expérimental
fd =
ued =
usd =
phid =
df = pd.DataFrame({'f [Hz]' : [,fd],
'ue [V]' : [,ued],
'us [V]' : [,usd],
'phi [deg]' : [,phid]},
index = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
columns = ['f [Hz]','ue [V]','us [V]','phi [deg]'])
df['omega [rad/s]'] = [df['f [Hz]'][i]*2*np.pi for i in df.index]
df['G [dB]'] = [20 * np.log10(df['us [V]'][i]/df['ue [V]'][i]) for i in df.index]
#print(df)
In [7]:
# Affichage
plt.subplot(211)
plt.title('Diagrammes de Bode')
plt.semilogx(df['omega [rad/s]'][len(df)], df['G [dB]'][len(df)],'og', markersize = 10)
plt.semilogx(df['omega [rad/s]'], df['G [dB]'],'+')
plt.semilogx(w, mag, lw=2) # Amplitude
#plt.xlabel('Pulsation (rad/s)')
plt.ylabel('Amplitude (dB)')
plt.grid(which='both')
plt.subplot(212)
plt.semilogx(df['omega [rad/s]'][len(df)], df['phi [deg]'][len(df)],'og', markersize = 10)
plt.semilogx(df['omega [rad/s]'],df['phi [deg]'],'+')
plt.semilogx(w, phase) # Phase
plt.xlabel('Pulsation (rad/s)')
plt.ylabel('Phase (deg)')
plt.grid(which='both')
plt.tight_layout() # Ajuster le placement des courbes
plt.show()
Diagramme de Bode du filtre de l'oscillateur de Wien¶
In [8]:
# Données
R = 10e3 # en ohm
#uR = 0.1e3 # en ohm
C = 100e-9 # en F
# uC = 0.1e-9 # en F
tau = R * C # en s
In [9]:
# Diagramme de Bode expériemental
# Direct
fd =
ued =
usd =
phid =
df = pd.DataFrame({'f [Hz]' : [,fd],
'ue [V]' : [,ued],
'us [V]' : [,usd],
'phi [deg]' : [,phid]},
index = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],
columns = ['f [Hz]','ue [V]','us [V]','phi [deg]'])
df['omega [rad/s]'] = [df['f [Hz]'][i]*2*np.pi for i in df.index]
df['G [dB]'] = [20 * np.log10(df['us [V]'][i]/df['ue [V]'][i]) for i in df.index]
In [10]:
# Fonction de transfert
ft = signal.lti([tau, 0], [tau**2, 3 * tau, 1])
# Calcul du diagramme de Bode
omega = np.logspace(0.8, 6, 1000)
w, mag, phase = signal.bode(ft, w=omega)
In [11]:
# Affichage
plt.subplot(211)
plt.title('Diagrammes de Bode')
plt.semilogx(df['omega [rad/s]'], df['G [dB]'],'+')
plt.semilogx(w, mag) # Amplitude
plt.ylabel('Amplitude (dB)')
plt.grid(which='both')
# Régressions linéaires
a_moy1, a_sig1, b_moy1, b_sig1 = montecarlo(df['omega [rad/s]'][:3], df['G [dB]'][:3], 0 , 0.1)
a_moy2, a_sig2, b_moy2, b_sig2 = montecarlo(df['omega [rad/s]'][7:11], df['G [dB]'][7:11], 0 , 0.1)
plt.semilogx(np.linspace(10, 10000), a_moy1 * np.log10(np.linspace(10, 10000)) + b_moy1, 'r--')
plt.semilogx(np.linspace(1000, 1000000), a_moy2 * np.log10(np.linspace(1000, 1000000)) + b_moy2, 'r--')
plt.subplot(212)
plt.semilogx(df['omega [rad/s]'],df['phi [deg]'],'+')
plt.semilogx(w, phase) # Phase
plt.xlabel('Pulsation (rad/s)')
plt.ylabel('Phase (deg)')
plt.grid(which='both')
plt.tight_layout() # Ajuster le placement des courbes
plt.show()
In [12]:
# Calcul de omega0_exp à l'intersection des deux régressions linéaires
omega0_exp = 10**((b_moy2 - b_moy1)/(a_moy1 - a_moy2))
uomega = omega0_exp*(np.sqrt((a_sig1/a_moy1)**2+(b_sig1/b_moy1)**2))
print("La fréquence propre expérimentale vaut: f0_exp =", np.round(omega0_exp/(2*np.pi)),'+/-',np.round(uomega,1), "Hz")
print("La fréquence propre théorique vaut: f0_th =", np.round(1/(tau * 2*np.pi)), "Hz")
La fréquence propre expérimentale vaut: f0_exp = +/- Hz La fréquence propre théorique vaut: f0_th = Hz
In [ ]:
Date de dernière mise à jour : 03/06/2025
