264 lines
6.7 KiB
Python
264 lines
6.7 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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## Part I (charge)
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##
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##
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## overdamped oscillation
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##
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R = ufloat(9.93e2, 0.05e2) # resistance
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L = ufloat(3.82e-3, 0.5e-3) # inductance
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C = ufloat(8.32e-6, 0.5e-6) # capacitance
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V0 = ufloat(10, 0) # generator peak tension
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nu = ufloat(2, 0) # generator frequency
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# time
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t = array(0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
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12, 13, 14, 15, 16, 17, 18, 19, 20)*1e-3
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# tension
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V = array(9.8, 9.56, 9.32, 9.09, 8.89, 7.87, 6.97, 6.17, 5.47,
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4.85, 4.3, 3.81, 3.37, 2.99, 2.65, 2.34, 2.09, 1.83,
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1.64, 1.46, 1.27, 1.14, 1.01, 0.9)
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# fit V(t)
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w0 = 1/um.sqrt(L*C)
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y = R/(2*L)
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a = V0
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b = um.sqrt(y**2 - w0**2)
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def over(t, a, b, y):
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return a * y/b * (np.exp(-(y-b)*t) - np.exp(-(y+b)*t))
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x = np.linspace(t.min(), t.max(), 500)
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f, p = curve(t, V, over, 0.01, guess=[V0.n, b.n, y.n])
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# χ² and compatibility tests
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alpha = chi_squared_fit(t, V, f, sigma=0.01, s=3)
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beta = [check_measures(*i) for i in zip([a, b, y], p)]
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print(mformat('''
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overdamped oscillation fit
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α: {}V, β₀={:.2f}
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β: {}Hz, β₁={:.2f}
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γ: {}Hz, β₂={:.2f}
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χ²: α={:.2f}, α>ε: {}
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''', p[0], beta[0]
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, p[1], beta[1]
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, p[2], beta[2]
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, alpha, alpha>epsilon))
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# plot V(t)
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plt.figure(1)
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plt.title('overdamped oscillations (charge)')
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plt.xlabel('time (ms)')
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plt.ylabel('tension (V)')
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plt.scatter(t*1e3, V, color='#5c6652')
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plt.plot(x*1e3, f(x), color='#718062')
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plt.show()
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##
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## underdamped oscillations
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##
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R = ufloat(100, 4) # resistance
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L = ufloat(3.82e-3, 0.5e-03) # inductance
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C = ufloat(44.5e-9, 0.5e-9) # capacitance
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V0 = ufloat(5, 0) # generator peak tension
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nu = ufloat(1, 0) # generator frequency
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# time
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t = np.append(np.arange(0, 100, 2), np.arange(104, 250, 4))*1e-6
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# tension
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V = array(0.285, 0.522, 0.712, 0.879, 1.040, 1.158, 1.229, 1.286,
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1.292, 1.256, 1.214, 1.117, 1.026, 0.900, 0.777, 0.605,
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0.437, 0.271, 0.121, -0.045, -0.196, -0.335, -0.453, -0.564,
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-0.636, -0.701, -0.758, -0.777, -0.770, -0.766, -0.715, -0.673,
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-0.602, -0.524, -0.450, -0.351, -0.258, -0.142, -0.062, 0.041,
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0.122, 0.211, 0.268, 0.341, 0.382, 0.426, 0.454, 0.466,
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0.471, 0.453, 0.366, 0.259, 0.135, 0.017, -0.081, -0.165,
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-0.233, -0.270, -0.287, -0.249, -0.217, -0.164, -0.078, -0.003,
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0.052, 0.109, 0.146, 0.147, 0.159, 0.146, 0.122, 0.100,
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0.050, -0.002, -0.034, -0.070, -0.073, -0.100, -0.115, -0.093,
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-0.061, -0.054, -0.023, -0.014, 0.023, 0.051, 0.059)
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# fit V(t)
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a = R*C*V0
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y = R/(2*L)
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w0 = 1/um.sqrt(L*C)
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w = um.sqrt(w0**2 - y**2)
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def under(t, a, w, y):
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return a * np.exp(-y*t) * (y*np.cos(w*t) + w*np.sin(w*t))
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x = np.linspace(0, t.max(), 500)
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f, p = curve(t, V, under, 0.01, guess=[a.n, w.n, y.n])
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# χ² and compatibility tests
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alpha = chi_squared_fit(t, V, f, sigma=0.01, s=3)
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beta = [check_measures(*i) for i in zip([a, w, y], p)]
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print(mformat('''
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underdamped oscillation fit
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α: {}Wb, β₀={:.2f}
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ω: {}Hz, β₁={:.2f}
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γ: {}Hz, β₂={:.2f}
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χ²: α={:.2f}, α>ε: {}
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''', p[0], beta[0]
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, p[1], beta[1]
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, p[2], beta[2]
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, alpha, alpha>epsilon))
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plt.figure(2)
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plt.title('underdamped oscillations (charge)')
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plt.xlabel('time (μs)')
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plt.ylabel('tension (V)')
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plt.scatter(t*1e6, V, s=12, color='#a17e3e')
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plt.plot(x*1e6, f(x), color='#c8b491')
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plt.show()
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##
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## critical dampening
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##
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R = ufloat(586, 4) # resistance
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L = ufloat(3.82e-3, 0.5e-3) # inductance
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C = ufloat(4.45e-8, 0.05e-8) # capacitance
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V0 = ufloat(2.5, 0) # generator peak tension
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# time
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t = array(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 18, 22, 26, 30, 34,
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38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90,
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94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138,
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142, 146, 150)*1e-6
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# tension
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V = array(0.007, 0.355, 0.654, 0.907, 1.120, 1.300, 1.437, 1.560,
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1.639, 1.726, 1.772, 1.821, 1.726, 1.557, 1.355, 1.145,
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0.954, 0.783, 0.642, 0.526, 0.411, 0.328, 0.254, 0.213,
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0.157, 0.125, 0.095, 0.069, 0.057, 0.040, 0.034, 0.026,
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0.025, 0.024, 0.013, 0.014, 0.003, 0.010, 0.010, 0.007,
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-0.001, 0.002, -0.004, -0.006, -0.008, 0.002)
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## fit as critical
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y = R/(2*L)
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a = R*C*V0
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def critic(t, a, y):
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return a * y**2 * t*np.exp(-y*t)
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x = np.linspace(0, t.max(), 500)
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f, p = curve(t, V, critic, 0.005, guess=[a.n, y.n])
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# χ² and compatibility tests
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alpha = chi_squared_fit(t, V, f, sigma=0.005, s=2)
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beta = [check_measures(*i) for i in zip([a, y], p)]
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print(mformat('''
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critically damped oscillation fit
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α: {}Wb, β₀={:.2f}
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γ: {}Hz, β₂={:.2f}
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χ²: α={:.2f}, α>ε: {}
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''', p[0], beta[0]
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, p[1], beta[1]
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, alpha, alpha>epsilon))
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# plot V(t)
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plt.figure(3)
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plt.subplot(3, 1, 1)
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plt.title('critically damped oscillations (charge)')
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plt.xlabel('time (μs)')
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plt.ylabel('tension (V)')
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plt.scatter(t*1e6, V, color='#43454f')
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plt.plot(x*1e6, f(x), color='#65788f')
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## fit as overdamped
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w0 = 1/um.sqrt(L*C)
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y = R/(2*L)
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a = V0
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b = um.sqrt(y**2 - w0**2)
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def over(t, a, b, y):
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return a * y/b * (np.exp(-(y-b)*t) - np.exp(-(y+b)*t))
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x = np.linspace(t.min(), t.max(), 500)
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f, p = curve(t, V, over, 0.01, guess=[V0.n, b.n, y.n])
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# χ² and compatibility tests
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alpha = chi_squared_fit(t, V, f, sigma=0.01, s=3)
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beta = [check_measures(*i) for i in zip([a, b, y], p)]
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print(mformat('''
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critically damped oscillation fit (as overdamped)
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α: {}V, β₀={:.2f}
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β: {}Hz, β₁={:.2f}
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γ: {}Hz, β₂={:.2f}
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χ²: α={:.2f}, α>ε: {}
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''', p[0], beta[0]
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, p[1], beta[1]
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, p[2], beta[2]
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, alpha, alpha>epsilon))
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# plot V(t)
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plt.subplot(3, 1, 2)
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plt.xlabel('time (ms)')
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plt.ylabel('tension (V)')
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plt.scatter(t*1e3, V, color='#5c6652')
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plt.plot(x*1e3, f(x), color='#718062')
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## fit as underdamped
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a = R*C*V0
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y = R/(2*L)
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w0 = 1/um.sqrt(L*C)
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w = um.sqrt(abs(w0**2 - y**2))
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def under(t, a, w, y):
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return a * np.exp(-y*t) * (y*np.cos(w*t) + w*np.sin(w*t))
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x = np.linspace(0, t.max(), 500)
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f, p = curve(t, V, under, 0.01, guess=[a.n, w.n, y.n])
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# χ² and compatibility tests
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alpha = chi_squared_fit(t, V, f, sigma=0.01, s=3)
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beta = [check_measures(*i) for i in zip([a, w, y], p)]
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print(mformat('''
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critically damped oscillation fit (as underdamped)
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α: {}Wb, β₀={:.2f}
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ω: {}Hz, β₁={:.2f}
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γ: {}Hz, β₂={:.2f}
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χ²: α={:.2f}, α>ε: {}
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''', p[0], beta[0]
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, p[1], beta[1]
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, p[2], beta[2]
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, alpha, alpha>epsilon))
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plt.subplot(3,1,3)
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plt.xlabel('time (μs)')
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plt.ylabel('tension (V)')
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plt.scatter(t*1e6, V, s=12, color='#a17e3e')
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plt.plot(x*1e6, f(x), color='#c8b491')
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plt.show()
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