24. Phénomènes de résonances dans différents domaines de la physique.
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RLC TENSION
In [12]:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
Création du tableau avec données expérimentales¶
In [13]:
df = pd.DataFrame({'R [ohm]' : [49.2,199,1290,698],
'L [H]' : [0.102,0.102,0.102,0.102],
'R_L [ohm]' : [118,118,118,118],
'C [nF]' : [101.5,101.5,101.5,101.5]},
index = [1,2,3,4],
columns = ['R [ohm]','L [H]','R_L [ohm]','C [nF]'])
df['C [F]'] = [df['C [nF]'][i]*1e-9 for i in df.index]
R_GBF = 50
df['fr [Hz]'] = []
print(df)
Calcul de R_tot et (fr et Q) théoriques¶
In [14]:
#Fonction pour caluler Q, fr et R_tot
def fr_th(L,C):
return 1/(np.sqrt(L*C)*2*np.pi)
df['fr_th [Hz]'] = [fr_th(df['L [H]'][i],df['C [F]'][i]) for i in df.index]
def R_tot(R,R_L,R_GBF): # R_tot prend en compte la résistance interne du GBF et celle de la bobine
return R_L+R+R_GBF
df['R_tot [ohm]'] = [R_tot(df['R [ohm]'][i],df['R_L [ohm]'][i],R_GBF) for i in df.index]
df['R_LG [ohm]'] = [df['R_L [ohm]'][i]+R_GBF for i in df.index]
def Q(R,L,C):
return (1/R)*np.sqrt(L/C)
df['Q'] = [Q(df['R_tot [ohm]'][i],df['L [H]'][i],df['C [F]'][i]) for i in df.index]
print(df)
Mesures de ue et ur pour différentes R¶
In [15]:
f = np.array([])
df['f [Hz]'] = [f for i in df.index]
ue_1 = np.array([])
uc_1 = np.array([])
phi_1 = np.array([])
ue_2 = np.array([])
uc_2 = np.array([])
phi_2 = np.array([])
ue_3 = np.array([])
uc_3 = np.array([])
phi_3 = np.array([])
# Mesure en direct
f_m = np.array([])
df['f [Hz]'][4] = f_m
ue_4 = np.array([])
uc_4 = np.array([])
phi_4 = np.array([])
df['ue [V]'] = [ue_1,ue_2,ue_3,ue_4]
df['uc [V]'] = [uc_1,uc_2,uc_3,uc_4]
df['phi [°]'] = [phi_1,phi_2,phi_3,phi_4]
C:\Users\Galy_\AppData\Local\Temp\ipykernel_9132\1813771301.py:18: SettingWithCopyWarning: A value is trying to be set on a copy of a slice from a DataFrame See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy df['f [Hz]'][4] = f_m
Calcul de ur théorique¶
In [16]:
df['f_th [Hz]'] = [np.linspace(400, 2600, 500) for i in df.index]
df['x'] = [df['f_th [Hz]'][i]/df['fr_th [Hz]'][i] for i in df.index]
df['ue_th [V]'] = 5
def u_th(ue,x,Q):
return(ue/(np.sqrt((1-x**2)**2+(x/Q)**2)))
df['u_th [V]'] = [u_th(df['ue_th [V]'][i],df['x'][i],df['Q'][i]) for i in df.index]
Affichage des résultats¶
In [17]:
df['col'] = ["b","r","g","m"]
plt.figure()
for i in df.index:
if i!=4:
plt.plot(df['f [Hz]'][i], df['uc [V]'][i], "+", color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm")
plt.plot(df['f_th [Hz]'][i], df['u_th [V]'][i], color = df['col'][i])
elif i==4:
plt.plot(df['f [Hz]'][i], df['uc [V]'][i], "o", color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm")
plt.plot(df['f_th [Hz]'][i], df['u_th [V]'][i], color = df['col'][i])
plt.xlabel('Fréquence [Hz]')
plt.ylabel('Tension aux bornes du condensateur [V]')
plt.title('Résonance en tension')
plt.legend()
plt.show()
Diagramme de Bode en gain¶
In [18]:
# Diagramme de Bode
df['x_m'] = [df['f [Hz]'][i]/df['fr [Hz]'][i] for i in df.index]
def Gain(uc,ue):
return 20*np.log10(uc/ue)
df['GdB [dB]'] = [Gain(df['uc [V]'][i],df['ue [V]'][i]) for i in df.index]
def Gain_th(x,Q):
return -20*np.log10(np.sqrt((1-x**2)**2+(x/Q)**2))
df['GdB_th [dB]'] = [Gain_th(df['x'][i],df['Q'][i]) for i in df.index]
In [19]:
plt.figure()
for i in df.index:
if i!=4:
plt.semilogx(df['x_m'][i],df['GdB [dB]'][i],"+",color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm") #pts expé
plt.semilogx(df['x'][i],df['GdB_th [dB]'][i],color = df['col'][i]) #theorie
elif i==4:
plt.semilogx(df['x_m'][i],df['GdB [dB]'][i],"o",color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm") #pts expé
plt.semilogx(df['x'][i],df['GdB_th [dB]'][i],color = df['col'][i]) #theorie
plt.xlabel("x = w/w0")
plt.ylabel("Gain [dB]")
plt.title("EDiagramme de Bode / résonance en tension")
plt.grid(which='both')
plt.legend()
plt.show()
Estimation de Q¶
In [20]:
plt.figure(0,figsize=(10,4)) #taille affichage
i = 1
plt.semilogx(df['f [Hz]'][i],df['GdB [dB]'][i],"b+",label="R1 = 50 ohm ") #pts expé
plt.semilogx(df['f_th [Hz]'][i],df['GdB_th [dB]'][i]) #courbe théorique
plt.axhline(max(df['GdB [dB]'][i])-3, color="grey", label="$GdB_{max}-3dB$") # tracé du gain efficace cad Gdb_max - 3 db
plt.axvline(1400, color="grey") # droite verticale gauche
plt.axvline(1700, color="grey") #droite verticale droite
plt.arrow(1400,0, 1700-1400,0, color="red")
plt.text(1500,-1, "$\Delta$f", color="red")
plt.xlabel("Fréquence [Hz]")
plt.ylabel("Gain [dB]")
plt.title("Evolution du gain en fonction de la fréquence")
plt.grid(which='both')
plt.legend()
plt.show()
df1 = 1700-1400 #delta f
f_max1 = df['f [Hz]'][1][np.argmax(df['uc [V]'][i])] #fréquence max
Q_exp1 = f_max1/df1 #calcul facteur qualité
udf = 50
ufmax = 20
uQ1 = Q_exp1*np.sqrt((udf/df1)**2+(ufmax/f_max1)**2) # erreur associé
print("Q_exp1 =",round(Q_exp1,2),"+/-",round(uQ1,2),"; Q_th1 =",round(df['Q'][i],2))
Q_exp1 = 5.0 +/- 0.84 ; Q_th1 = 4.62
RLC COURANT
In [131]:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
Création du tableau avec données expérimentales¶
In [189]:
df = pd.DataFrame({'R [ohm]' : [50,199,1290,698],
'L [H]' : [0.104,0.104,0.104,0.104],
'R_L [ohm]' : [118,118,118,118],
'C [nF]' : [100,100,100,100]},
index = [1,2,3,4],
columns = ['R [ohm]','L [H]','R_L [ohm]','C [nF]'])
df['C [F]'] = [df['C [nF]'][i]*1e-9 for i in df.index]
R_GBF = 50
df['fr [Hz]'] = []
print(df)
R [ohm] L [H] R_L [ohm] C [nF] C [F] fr [Hz] 1 50 0.104 118 100 1.000000e-07 1570 2 199 0.104 118 100 1.000000e-07 1570 3 1290 0.104 118 100 1.000000e-07 1570 4 698 0.104 118 100 1.000000e-07 1570
Calcul de R_tot et (fr et Q) théoriques¶
In [190]:
#Fonction pour caluler Q, fr et R_tot
def fr_th(L,C):
return 1/(np.sqrt(L*C)*2*np.pi)
df['fr_th [Hz]'] = [fr_th(df['L [H]'][i],df['C [F]'][i]) for i in df.index]
def R_tot(R,R_L,R_GBF): # R_tot prend en compte la résistance interne du GBF et celle de la bobine
return R_L+R+R_GBF
df['R_tot [ohm]'] = [R_tot(df['R [ohm]'][i],df['R_L [ohm]'][i],R_GBF) for i in df.index]
df['R_LG [ohm]'] = [df['R_L [ohm]'][i]+R_GBF for i in df.index]
def Q(R,L,C):
return (1/R)*np.sqrt(L/C)
df['Q'] = [Q(df['R_tot [ohm]'][i],df['L [H]'][i],df['C [F]'][i]) for i in df.index]
Mesures de ue et uc pour différentes R¶
In [197]:
f = np.array([])
df['f [Hz]'] = [f for i in df.index]
ue_1 = np.array([])
ur_1 = np.array([])
phi_1 = np.array([])
ue_2 = np.array([])
ur_2 = np.array([])
phi_2 = np.array([])
ue_3 = np.array([])
ur_3 = np.array([])
phi_3 = np.array([])
ue_4 = np.array([])
ur_4 = np.array([])
phi_4 = np.array([])
df['ue [V]'] = [ue_1,ue_2,ue_3,ue_4]
df['ur [V]'] = [ur_1,ur_2,ur_3,ur_4]
df['phi [°]'] = [phi_1,phi_2,phi_3,phi_4]
Calcul de ur théorique¶
In [198]:
df['f_th [Hz]'] = [np.linspace(400, 2600, 500) for i in df.index]
df['x'] = [df['f_th [Hz]'][i]/df['fr_th [Hz]'][i] for i in df.index]
df['ue_th [V]'] = 5
def u_th(ue,x,Q):
return(ue/(np.sqrt(1+Q**2*(x-(1/x))**2)))
df['u_th [V]'] = [u_th(df['ue_th [V]'][i],df['x'][i],df['Q'][i]) for i in df.index]
def u_th2(ue,R,C,L,R2,f):
w = 2*np.pi*f
return ue*R*C*w/(np.sqrt((1-L*C*w**2)**2+(C*w*(R+R2))**2))
df['u_th2'] = [u_th2(df['ue_th [V]'][i],df['R [ohm]'][i],df['C [F]'][i],df['L [H]'][i],df['R_LG [ohm]'][i],df['f_th [Hz]'][i]) for i in df.index]
Affichage des résultats¶
In [199]:
df['col'] = ["b","r","g","m"]
plt.figure()
for i in df.index:
plt.plot(df['f [Hz]'][i], df['ur [V]'][i], "+", color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm")##
# plt.plot(df['f_th [Hz]'][i], df['u_th [V]'][i], color = df['col'][i])
plt.plot(df['f_th [Hz]'][i], df['u_th2'][i], color = df['col'][i])
plt.xlabel('Fréquence [Hz]')
plt.ylabel('Tension aux bornes de la résistance [V]')
plt.title('Résonance en courant')
plt.legend()
plt.show()
Diagramme de Bode en gain¶
In [200]:
# Diagramme de Bode
df['x_m'] = [df['f [Hz]'][i]/df['fr [Hz]'][i] for i in df.index]
def Gain(ur,ue):
return 20*np.log10(ur/ue)
df['GdB [dB]'] = [Gain(df['ur [V]'][i],df['ue [V]'][i]) for i in df.index]
def Gain_th(x,Q):
return -20*np.log10((np.sqrt(1+Q**2*(x-(1/x))**2)))
df['GdB_th [dB]'] = [Gain_th(df['x'][i],df['Q'][i]) for i in df.index]
def Gain_th2(R,C,L,R2,f):
w = 2*np.pi*f
return 20*np.log10(R*C*w/(np.sqrt((1-L*C*w**2)**2+(C*w*(R+R2))**2)))
df['GdB_th2'] = [Gain_th2(df['R [ohm]'][i],df['C [F]'][i],df['L [H]'][i],df['R_L [ohm]'][i],df['f_th [Hz]'][i]) for i in df.index]
In [201]:
plt.figure()
for i in df.index:
plt.semilogx(df['x_m'][i],df['GdB [dB]'][i],"+",color = df['col'][i], label="R = "+str(df['R [ohm]'][i])+" ohm") #pts expé
#plt.semilogx(df['x'][i],df['GdB_th [dB]'][i],color = df['col'][i]) #theorie
plt.semilogx(df['x'][i],df['GdB_th2'][i],color = df['col'][i]) #theorie
plt.xlabel("x = w/w0")
plt.ylabel("Gain [dB]")
plt.title("Diagramme de Bode / résonance en courant")
plt.grid(which='both')
plt.legend()
plt.show()
Estimation de Q¶
In [196]:
plt.figure(0,figsize=(10,4)) #taille affichage
i = 1
plt.semilogx(df['f [Hz]'][i],df['GdB [dB]'][i],"b+",label="R1 = 50 ohm ") #pts expé
#plt.semilogx(df['f_th [Hz]'][i],df['GdB_th [dB]'][i]) #courbe théorique
plt.semilogx(df['f_th [Hz]'][i],df['GdB_th2'][i]) #courbe théorique
plt.axhline(max(df['GdB [dB]'][i])-3, color="grey", label="$GdB_{max}-3dB$") # tracé du gain efficace cad Gdb_max - 3 db
plt.axvline(1400, color="grey") # droite verticale gauche
plt.axvline(1700, color="grey") #droite verticale droite
plt.arrow(1400,-22, 1700-1400,0, color="red")
plt.text(1500,-24, "$\Delta$f", color="red")
plt.xlabel("Fréquence [Hz]")
plt.ylabel("Gain [dB]")
plt.title("Evolution du gain en fonction de la fréquence")
plt.grid(which='both')
plt.legend()
plt.show()
df1 = 1700-1400 #delta f
f_max1 = df['f [Hz]'][1][np.argmax(df['ur [V]'][i])] #fréquence max
Q_exp1 = f_max1/df1 #calcul facteur qualité
udf = 50
ufmax = 20
uQ1 = Q_exp1*np.sqrt((udf/df1)**2+(ufmax/f_max1)**2) # erreur associé
print("Q_exp1 =",round(Q_exp1,2),"+/-",round(uQ1,2),"; Q_th1 =",round(df['Q'][i],2))
Q_exp1 = 5.17 +/- 0.86 ; Q_th1 = 4.68
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Date de dernière mise à jour : 07/06/2025