275 lines
7.2 KiB
Python
275 lines
7.2 KiB
Python
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# 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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##
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## Impedance of a resistor
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## (all SI units)
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R = ufloat(996, 4) # resistor (for oscilloscope)
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# frequency
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nu = array(60, 100, 150, 200, 300, 500, 750, 1e3, 1.5e3, 2e3, 3e3,
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5e3, 8e3, 12e3, 15e3, 20e3, 25e3, 40e3, 50e3, 60e3, 75e3,
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100e3, 150e3, 200e3, 250e3, 300e3, 350e3)
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# tension drop (via voltmeter)
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Vm = array(1.564, 1.564, 1.562, 1.558, 1.563, 1.552, 1.536, 1.518,
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1.470, 1.417, 1.303, 1.096, 0.878, 0.715, 0.644, 0.575,
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0.533, 0.470, 0.445, 0.423, 0.395, 0.349, 0.257, 0.169,
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0.085, 0.038, 0.020)*np.sqrt(2)
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# current (via amperometer)
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Im = array(1.663, 1.664, 1.663, 1.664, 1.664, 1.664, 1.664, 1.664,
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1.664, 1.664, 1.664, 1.666, 1.671, 1.679, 1.687, 1.702,
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1.721, 1.798, 1.865, 1.943, 2.076, 2.327, 2.817, 3.143,
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3.255, 3.294, 3.301)*1e-3*np.sqrt(2)
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# tension input - ground (via oscilloscope)
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Va = array(9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6,
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9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6,
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9.6, 9.6, 9.6, 9.6, 9.6, 9.6, 9.6)/2
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# tension output - ground (via oscilloscope)
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Vb = array(4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8,
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4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8,
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4.8, 4.8, 4.8, 4.8, 4.8, 4.8, 4.8)/2
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# add uncertainties
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Vb, Va = ufloat(Vb[0], 0.01), ufloat(Va[0], 0.01)
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Io = Vb/R # current (via oscilloscope)
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Vo = Va - Vb # tension drop (via oscilloscope)
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Z = Vo/Io # impedance
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plt.figure(1)
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plt.clf()
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plt.xlabel("frequency (Hz)")
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plt.ylabel("voltage voltmeter / oscilloscope")
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plt.semilogx(nu, Vm/Vo.n, 'o-', color='#bb3e5f', markersize=4.5)
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plt.show()
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plt.figure(2)
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plt.clf()
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plt.xlabel("frequency (Hz)")
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plt.ylabel("current amperometer / oscilloscope")
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plt.semilogx(nu, Io.n/Im, 'o-', color="#56ad5e", markersize=4.5)
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plt.show()
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plt.figure(3)
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plt.clf()
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plt.title('impedance (purely resistive circuit)')
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plt.xlabel('frequency (Hz)')
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plt.ylabel('magnitude (kΩ)')
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plt.semilogx(nu, nu/nu*Z.n/1e3, 'o-', color="#d0aa23", markersize=4.5)
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plt.show()
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alpha = check_measures(Z, R)
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print(mformat('''
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Z: {:.3}Ω
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R: {:.3}Ω
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compatibility test:
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α={:.2f}, α>ε: {}
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''', Z, R, alpha, alpha>epsilon))
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##
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## Impedance of a capacitor
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## (all SI units)
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C = ufloat(10.74e-9, 1e-9) # capacitor
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R = ufloat(996, 4) # resistor
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# frequency (Hz)
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nu = array(60, 75, 100, 150, 200, 300, 400, 500, 600, 700, 800, 1000, 1200,
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1300, 1400, 1500, 1800, 2500, 5000, 10e3, 50e3, 100e3, 150e3,
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200e3, 300e3, 350e3)
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# V input
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Va = array(10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2,
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10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2, 10.2,
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10.2, 10.2, 10.2, 10.2)/2
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# V output
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Vb = array(45.0e-3, 55.2e-3, 74e-3, 106e-3, 140e-3, 212e-3, 276e-3,
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344e-3, 460e-3, 520e-3, 580e-3, 700e-3, 860e-3, 920e-3,
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1.05, 1.16, 1.41, 1.88, 3.60, 6.15, 9.89, 10.09, 10.16,
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10.16, 10.17, 10.17)/2
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# time offset Va - Vb
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Oab = array(4.123, 3.300, 2.480, 1.650, 1.230, 0.820, 0.611, 0.490,
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0.404, 0.344, 0.302, 0.240, 0.196, 0.181, 0.167, 0.155,
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0.128, 0.089, 0.040, 0.016, 0.95e-3, 0.23e-3, 0.10e-3,
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0.05e-3, 0.025e-3, 0.020e-3)*(-1e-3)
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# time offset I - V
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Oiv = array(4.200e-3, 3.320e-3, 2.500e-3, 1.645e-3, 1.250e-3, 830e-6, 635e-6,
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498e-6, 420e-6, 350e-6, 310e-6, 250e-6, 208e-6, 192e-6,
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176e-6, 166e-6, 138e-6, 100e-6, 50e-6, 25e-6, 4.91e-6,
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2.49e-6, 1.68e-6, 1230e-9, 826e-9, 720e-9)*(-1)
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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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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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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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# 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 (capacitive 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=1/ν, k=1/2πC
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k = simple_linear(1/nu, abs(Z), sigmaZ)
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Co = 1/(2*np.pi*k)
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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.ylim(-1.8, -1)
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plt.semilogx(nu, Fiv, 'o', color="#36913d", markersize=4.5)
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plt.semilogx(x, x/x * -np.pi/2, color="#589f22")
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plt.show()
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phi = sample(Fiv).val()
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alpha = check_measures(phi, ufloat(-np.pi/2, 0))
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print(mformat('''
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φ: {} rad
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-π/2: {:.4} rad
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compatibility test:
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α={:.2f}, α>ε: {}
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''', phi, -np.pi/2, alpha, alpha>epsilon))
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alpha = check_measures(C, Co)
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beta = chi_squared_fit(nu, abs(Z), f, sigmaZ)
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print(mformat('''
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k: {}
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C: {} F
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Cₒ: {} F
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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, C, Co,
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alpha, alpha>epsilon,
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beta, beta>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('transmission function 1')
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plt.ylabel('amplitude (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=1/ν, k=1/2πRC
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k = simple_linear(1/nu, abs(H1), sigmaH1)
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RCo = 1/(2*np.pi*k)
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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.semilogx(x, -np.pi/2+np.arctan(2*np.pi*x*R.n*C.n))
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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('transmission function 2')
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plt.ylabel('amplitude (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)
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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, -np.pi/2+np.arctan(2*np.pi*x*R.n*C.n))
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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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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:
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β={:.2f}, β>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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