Active Filter Design — Butterworth & Chebyshev Approximations

Active Filter Design — Butterworth & Chebyshev Approximations

1. General 2nd-Order Transfer Function

Every 2nd-order filter section (biquad) has the form:

$$H(s) = \frac{K \omega_0^2}{s^2 + (\omega_0/Q)s + \omega_0^2} \quad \text{(low-pass)}$$

where $\omega_0 = 2\pi f_0$ is the natural frequency, $Q$ is the quality factor, and $K$ is the DC gain. The $-3$ dB frequency of a 2nd-order Butterworth section ($Q = 1/\sqrt{2} = 0.7071$) equals $\omega_0$. Higher $Q$ produces peaking in the passband.

2. Butterworth Prototype

An $n$th-order Butterworth low-pass filter has the magnitude response:

$$|H(j\Omega)|^2 = \frac{1}{1 + \Omega^{2n}}$$

where $\Omega = f/f_0$ is the normalised frequency. The attenuation at $\Omega = 1$ is always $-3$ dB regardless of order. At $\Omega = \Omega_s$ (stopband edge), attenuation = $-10\log_{10}(1 + \Omega_s^{2n})$ dB. Required order to achieve $A_s$ dB attenuation at $\Omega_s$:

$$n \geq \frac{A_s/10 - \log_{10}(1)}{\log_{10}(\Omega_s^2)} = \frac{A_s - 3}{20 \log_{10}(\Omega_s)}$$

Wait — more precisely: $n \geq \log_{10}(10^{A_s/10}-1) / (2\log_{10}(\Omega_s))$.

Butterworth poles lie on the unit circle at angles $\theta_k = 90° + (2k-1)180°/n$ for $k = 1, \ldots, n$. For $n=4$: poles at $\pm 22.5°, \pm 67.5°$ from the $j\omega$ axis, giving two biquad sections with $Q_1 = 1/(2\sin 22.5°) = 1.307$ and $Q_2 = 1/(2\sin 67.5°) = 0.541$.

3. Sallen–Key Low-Pass Topology

The unity-gain Sallen–Key second-order LP section uses two resistors $R_1, R_2$ and two capacitors $C_1, C_2$:

$$\omega_0 = \frac{1}{\sqrt{R_1 R_2 C_1 C_2}}, \qquad Q = \frac{\sqrt{R_1 R_2 C_1 C_2}}{R_1 C_2 + R_2 C_2}$$

Design procedure for equal-capacitor ($C_1 = C_2 = C$) simplification: $R_1 = 1/(2\pi f_0 C) \cdot 1/Q$, $R_2 = 1/(2\pi f_0 C) \cdot Q$. Choose $C$ from preferred values (e.g., 10 nF at audio, 1 nF at RF), then compute $R_1, R_2$.

4. Component Sensitivity

The Sallen–Key topology has low component sensitivity: $S^{\omega_0}_{R_1} = -0.5$ (a 1% change in $R_1$ causes a 0.5% change in $\omega_0$). The MFB topology has better $Q$ stability but inverts the signal and requires tighter component matching.