一阶 RC 低通滤波器的传递函数为
H(j\omega) = \frac{1}{1 + j\omega RC}
\tag{1}
由 式 1，|H| 在截止角频率 \omega_c = 1/RC 处降到 1/\sqrt{2}，即 -3\ \mathrm{dB}；\omega \gg \omega_c 时，幅值以 -20\ \mathrm{dB/dec} 下降，相位趋于 -90^\circ [1]。图 1 取 R = 1\ \mathrm{k\Omega}、C = 159\ \mathrm{nF}，对应 f_c = 1/(2\pi RC) \approx 1\ \mathrm{kHz}。
代码
import matplotlib.pyplot as plt
import numpy as np
R, C = 1e3, 159e-9
f = np.logspace(1, 6, 500)
H = 1 / (1 + 2j * np.pi * f * R * C)
fig, (mag, phase) = plt.subplots(2, sharex=True)
mag.semilogx(f, 20 * np.log10(abs(H)))
mag.set_ylabel("|H| (dB)")
phase.semilogx(f, np.degrees(np.angle(H)))
phase.set_ylabel("phase (deg)")
phase.set_xlabel("f (Hz)")
plt.show()
参考文献
[1]
A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1997.
