爱你我为冷风吹～吹到 GM4500 上面变成了有源滤波器

Vin ──R1──┬── R2 ──┬──→ 运放 + 输入
          |        |
         C1       C2
          |        |
         GND      GND

运放配置为非反相（或跟随）

Design summary:
Stage f0 = 100.0 kHz
RC fc    = 159.2 kHz (R=1k, C=1nF)
4th-order Qs: 0.541 1.306  Gains: 1.152 2.234
6th-order Qs: 0.518 0.707 1.932  Gains: 1.069 1.586 2.482

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

# ---------- Helper functions ----------
def lp2_stage_tf(f0_hz: float, Q: float, G: float = 1.0):
    """2nd-order low-pass stage: H(s)=G*w0^2/(s^2 + (w0/Q)s + w0^2)"""
    w0 = 2 * np.pi * f0_hz
    num = [G * w0**2]
    den = [1.0, (w0 / Q), w0**2]
    return signal.TransferFunction(num, den)

def lp1_rc_tf(fc_hz: float):
    """1st-order RC low-pass: H(s)=1/(1 + s/wc)"""
    wc = 2 * np.pi * fc_hz
    return signal.TransferFunction([wc], [1.0, wc])

def cascade(tfs):
    """Cascade a list of TransferFunction objects (continuous-time)."""
    num = np.array([1.0])
    den = np.array([1.0])
    for tf in tfs:
        num = np.polymul(num, tf.num)
        den = np.polymul(den, tf.den)
    return signal.TransferFunction(num, den)

def bode_mag_phase(tf, w):
    w_out, mag_db, phase_deg = signal.bode(tf, w=w)
    return w_out/(2*np.pi), mag_db, phase_deg

# ---------- Filters to visualize ----------
f0 = 100e3  # stage natural frequency (Hz)

# 2nd-order Butterworth (Q=0.707), SK equal-RC implies gain G=1.586
Q_2 = 0.707
G_2 = 1.586
H_2nd = lp2_stage_tf(f0, Q_2, G_2)

# 4th-order Butterworth via two biquads
Q4_1, Q4_2 = 0.541, 1.306
G4_1, G4_2 = 3 - 1/Q4_1, 3 - 1/Q4_2
H_4th = cascade([lp2_stage_tf(f0, Q4_1, G4_1), lp2_stage_tf(f0, Q4_2, G4_2)])

# 6th-order Butterworth via three biquads
Q6_1, Q6_2, Q6_3 = 0.518, 0.707, 1.932
G6_1, G6_2, G6_3 = 3 - 1/Q6_1, 3 - 1/Q6_2, 3 - 1/Q6_3
H_6th = cascade([lp2_stage_tf(f0, Q6_1, G6_1), lp2_stage_tf(f0, Q6_2, G6_2), lp2_stage_tf(f0, Q6_3, G6_3)])

# Front-end RC anti-alias filter (R=1k, C=1nF)
fc_rc = 1 / (2 * np.pi * 1e3 * 1e-9)
H_RC = lp1_rc_tf(fc_rc)

# Combined
H_RC_4th = cascade([H_RC, H_4th])
H_RC_6th = cascade([H_RC, H_6th])

# ---------- Frequency axis ----------
f = np.logspace(2, 8, 2000)  # 100 Hz to 100 MHz
w = 2 * np.pi * f

# ---------- Magnitude ----------
plt.figure()
for name, tf in [
    ("1st-order RC (fc≈159 kHz)", H_RC),
    ("2nd-order (Q=0.707)", H_2nd),
    ("4th-order Butterworth", H_4th),
    ("6th-order Butterworth", H_6th),
    ("RC + 4th-order", H_RC_4th),
    ("RC + 6th-order", H_RC_6th),
]:
    ff, mag, _ = bode_mag_phase(tf, w)
    plt.semilogx(ff, mag, label=name)
plt.xlabel("Frequency (Hz)")
plt.ylabel("Magnitude (dB)")
plt.title("Magnitude Response Comparison")
plt.grid(True, which="both")
plt.legend()
plt.show()

# ---------- Phase ----------
plt.figure()
for name, tf in [
    ("1st-order RC (fc≈159 kHz)", H_RC),
    ("2nd-order (Q=0.707)", H_2nd),
    ("4th-order Butterworth", H_4th),
    ("6th-order Butterworth", H_6th),
    ("RC + 4th-order", H_RC_4th),
    ("RC + 6th-order", H_RC_6th),
]:
    ff, _, ph = bode_mag_phase(tf, w)
    plt.semilogx(ff, ph, label=name)
plt.xlabel("Frequency (Hz)")
plt.ylabel("Phase (deg)")
plt.title("Phase Response Comparison")
plt.grid(True, which="both")
plt.legend()
plt.show()

# ---------- Step response ----------
t = np.linspace(0, 20e-6, 4000)  # 0..20 us
plt.figure()
for name, tf in [
    ("2nd-order (Q=0.707)", H_2nd),
    ("4th-order Butterworth", H_4th),
    ("6th-order Butterworth", H_6th),
    ("RC + 4th-order", H_RC_4th),
    ("RC + 6th-order", H_RC_6th),
]:
    tt, y = signal.step(tf, T=t)
    plt.plot(tt*1e6, y, label=name)
plt.xlabel("Time (µs)")
plt.ylabel("Step Response (V/Vstep)")
plt.title("Step Response Comparison")
plt.grid(True)
plt.legend()
plt.show()

print("Design summary:")
print(f"Stage f0 = {f0/1e3:.1f} kHz")
print(f"RC fc    = {fc_rc/1e3:.1f} kHz (R=1k, C=1nF)")
print("4th-order Qs:", Q4_1, Q4_2, " Gains:", round(G4_1,3), round(G4_2,3))
print("6th-order Qs:", Q6_1, Q6_2, Q6_3, " Gains:", round(G6_1,3), round(G6_2,3), round(G6_3,3))