189 lines
4.5 KiB
Python
189 lines
4.5 KiB
Python
# coding: utf-8
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from __future__ import print_function, division, unicode_literals
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import numpy as np
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import uncertainties.umath as um
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import matplotlib.pyplot as plt
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from lab import *
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## Impedence of an inductor (I)
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## (all SI units)
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R = ufloat(996, 4) # resistor
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L = ufloat(0.014561, 0.000009)
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C = ufloat(10e-12, 1e-12)
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# frequency
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nu = array( 2e3, 10e3, 25e3, 50e3, 75e3, 100e3, 125e3, 150e3, 175e3, 200e3,
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225e3, 250e3, 260e3, 270e3, 275e3, 285e3, 300e3, 350e3, 375e3)
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# V input
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Va = array(10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0,
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10.0, 10.0, 8.2, 10.0, 10.0, 8.20, 10.0, 10.0, 10.0)/2
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# V output
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Vb = array( 10.0, 9.4, 7.8, 5.0, 3.28, 2.20, 1.44, 0.96, 0.56, 0.32, 0.144,
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0.085, 0.0396, 0.069, 75.4e-3, 0.144, 0.194, 0.320, 0.360)/2
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# time offset Va - Vb
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Oab = array( 12e-6, 6.0e-6, 5.2e-6, 4.2e-6, 3.5e-6, 3.08e-6, 2.64e-6, 2.36e-6,
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2.08e-6, 1.88e-6, 1.71e-6, 260e-9, 260e-9, 280e-9, 280e-9, 300e-9,
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250e-9, 220e-9, 180e-9)
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# time offset I - V
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Oiv = array( 110e-6, 26.0e-6, 11.2e-6, 5.9e-6, 4.2e-6, 3.36e-6, 2.80e-6, 2.40e-6,
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2.12e-6, 1.90e-6, 1.74e-6, 280e-9, 260e-9, 270e-9, 280e-9, 280e-9,
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260e-9, 220e-9, 184e-9)
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om = 2*np.pi*nu # angular frequency
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I = Vb/R.n # output current
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Vab = Va - Vb # tension drop
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Fab = om * Oab # phase difference Vᵢ - Vₒ
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Fiv = om * Oiv # phase difference I - Vₒ
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Z = Vab/I * np.exp(1j*Fiv) # impedance
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H1 = Vab/Vb * np.exp(1j*Fab) # transfer function ΔV→Vb
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H2 = Vb/Va * np.exp(1j*Fab) # transfer function Va→Vb
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# estimate uncertainties
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Evb, Eva = ufloat(Vb[0], 1e-3), ufloat(Va[0], 0.1)
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sigmaF = (om[0]*ufloat(Oiv[0], 3e-5)).s
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sigmaZ = (R*(Eva - Evb)/Evb).s
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sigmaH1 = ((Eva - Evb)/Evb).s
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sigmaH2 = ((Evb - Eva)/Evb).s
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# plot and fit Z(ν)
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plt.figure(4)
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plt.clf()
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# magnitude
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plt.subplot(2, 1, 1)
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plt.title('impedance (RL circuit)')
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plt.ylabel('magnitude (kΩ)')
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plt.semilogx(nu, abs(Z)/1e3, 'o', color="#36913d", markersize=4.5)
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# fit Y=kX where Y=|Z|, X=ν, k=2πL
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k = simple_linear(nu, abs(Z), sigmaZ)
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L = k/(2*np.pi)
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f = lambda x: k.n*x
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x = np.arange(nu.min()-10, nu.max(), 10)
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plt.semilogx(x, f(x)/1e3, color='#589f22')
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# phase
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plt.subplot(2, 1, 2)
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plt.xlabel('frequency (Hz)')
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plt.ylabel('phase (rad)')
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plt.semilogx(nu, Fiv, 'o', color="#36913d", markersize=4.5)
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plt.show()
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alpha = chi_squared_fit(nu, abs(Z), f, sigmaZ)
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print(mformat('''
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k: {}
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L: {} H
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χ² test:
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α={:.2f}, α>ε: {}
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''', k, L,
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alpha, alpha>epsilon))
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# plot, fit H₁(ν)
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plt.figure(5)
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plt.clf()
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# magnitude
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plt.subplot(2,1,1)
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plt.title('transfer function 1')
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plt.ylabel('magnitude (Vout-Vin / Vout)')
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plt.semilogx(nu, abs(H1), 'o', color="#9b2e83", markersize=4.5)
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# fit Y=kX where Y=|H1|, X=ν, k=2πL/R
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k = simple_linear(nu, abs(H1), sigmaH1)
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L = k*R/(2*np.pi)
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f = lambda x: k.n/x
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x = np.arange(nu.min()-10, nu.max(), 10)
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plt.semilogx(x, f(x), color='#9b2e83')
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# phase
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plt.subplot(2,1,2)
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plt.xlabel('frequency (Hz)')
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plt.ylabel('phase (rad)')
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plt.semilogx(nu, Fab, 'o', color="#3a44ad", markersize=4.5)
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plt.show()
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alpha = check_measures(R*C, RCo)
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beta = chi_squared_fit(nu, abs(H1), f, sigmaH1)
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print(mformat('''
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k: {} Hz
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RC: {} s
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RCₒ: {} s
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compatibility test:
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α={:.2f}, α>ε: {}
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χ² test:
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β={:.2f}, β>ε: {}
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''', k, R*C, RCo,
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alpha, alpha>epsilon,
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beta, beta>epsilon))
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# plot, fit H₂(ν)
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plt.figure(6)
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plt.clf()
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# magnitude
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plt.subplot(2,1,1)
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plt.title('transfer function 2')
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plt.ylabel('magnitude (Vout / Vin)')
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plt.semilogx(nu, abs(H2), 'o', color="#9b2e83", markersize=4.5)
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# fit Y=a+bX where Y=1/|H₂|², X=1/ν², a=1, b=1/(2πRC)²
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a,b = linear(1/nu**2, 1/abs(H2)**2, 0.01)
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RCo = 1/(2*np.pi*um.sqrt(b))
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f = lambda x: 1/np.sqrt(1 + b.n/x**2) # magnitude
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g = lambda x: -np.pi/2 + np.arctan(2*np.pi*x*R.n*C.n) # phase
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x = np.arange(nu.min()-10, nu.max(), 10)
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plt.semilogx(x, f(x), color='#9b2e83')
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# phase
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plt.subplot(2,1,2)
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plt.xlabel('frequency (Hz)')
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plt.ylabel('phase (rad)')
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plt.semilogx(nu, Fab, 'o', color="#3a44ad", markersize=4.5)
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plt.semilogx(x, g(x))
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plt.show()
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alpha = check_measures(R*C, RCo)
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beta = chi_squared_fit(nu, abs(H2), f, sigmaH2)
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gamma = chi_squared_fit(nu, Fab, g, sigmaF)
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print(mformat('''
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b: {}
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RC: {} s
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RCₒ: {} s
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compatibility test:
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α={:.2f}, α>ε: {}
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χ² test (magnitude):
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β={:.2f}, β>ε: {}
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χ² test (phase):
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γ={:.2f}, γ>ε: {}
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''', b, R*C, RCo,
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alpha, alpha>epsilon,
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beta, beta>epsilon,
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gamma, gamma>epsilon))
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