Active Filters

Why Active Filters?

1. Gain: Can amplify passband signals (impossible with passive) 2. Isolation: High $Z_{in}$ and Low $Z_{out}$ prevent loading effects 3. No Inductors: Uses only R and C (smaller, cheaper, no magnetic coupling) 4. Easy Cascading: Multiple stages without impedance matching 5. Tuning: Electronically adjustable using voltage-controlled elements

Filter Classifications

By Frequency Response:

• Low-Pass (LPF): Passes DC to $f_c$, rejects above
• High-Pass (HPF): Rejects DC to $f_c$, passes above
• Band-Pass (BPF): Passes only frequencies between $f_1$ and $f_2$
• Band-Stop (Notch): Rejects frequencies between $f_1$ and $f_2$
• All-Pass: Constant magnitude, variable phase (delay equalization)

By Approximation:

• Butterworth: Maximally flat passband, -20n dB/decade rolloff
• Chebyshev: Ripple in passband, steeper rolloff than Butterworth
• Bessel: Maximally flat group delay (linear phase, best pulse response)
• Elliptic (Cauer): Ripple in both passband and stopband, steepest rolloff

First-Order Active Filters

Low-Pass Filter

Non-Inverting: $$H(s) = \frac{K}{1 + s/\omega_c}$$

• Cutoff: $f_c = \frac{1}{2\pi RC}$
• DC Gain: $K = 1 + R_2/R_1$
• Roll-off: -20dB/decade (-6dB/octave)

High-Pass Filter

Non-Inverting: $$H(s) = \frac{Ks}{s + \omega_c}$$

• Same component swap: C in series, R to ground

Second-Order Active Filters

Sallen-Key Topology

Most popular 2nd-order configuration (non-inverting, unity or low gain).

Advantages:

• Minimum component count
• Low sensitivity to component tolerances
• Easy to cascade for higher orders

Transfer Function (Low-Pass): $$H(s) = \frac{K\omega_0^2}{s^2 + \frac{\omega_0}{Q}s + \omega_0^2}$$

Where:

• $\omega_0 = 2\pi f_0$: Natural frequency (cutoff)
• $Q$: Quality factor (damping)
• $K$: DC gain (typically 1 for Sallen-Key)

Q Factor and Filter Response

Butterworth (Maximally Flat):

• 2nd order: $Q = 0.707$
• 4th order: $Q_1 = 0.541$, $Q_2 = 1.307$ (cascade two stages)
• 6th order: Three cascaded stages with specific Q values

Chebyshev (0.5dB ripple):

• 2nd order: $Q = 0.957$
• Steeper rolloff than Butterworth

Multiple Feedback (MFB) Topology

Alternative to Sallen-Key, inverting configuration.

Advantages:

• Better high-frequency performance
• Lower sensitivity to op-amp GBP

Design Procedure (Sallen-Key Butterworth LPF)

Given: $f_c = 1$ kHz, 2nd order Butterworth

Step 1: Calculate $\omega_0$ $$\omega_0 = 2\pi f_c = 2\pi(1000) = 6283 \text{ rad/s}$$

Step 2: Set $Q = 0.707$ (Butterworth)

Step 3: Choose capacitor values (equal for simplicity) $$C_1 = C_2 = 10 \text{ nF}$$

Step 4: Calculate resistors For equal capacitors and unity gain Sallen-Key: $$R_1 = R_2 = \frac{1}{\omega_0 C \sqrt{2}} = \frac{1}{6283 \times 10^{-8} \times 1.414} = 11.25 \text{ kΩ}$$

Use standard value: 11 kΩ or 12 kΩ

Step 5: Set gain (unity gain buffer in feedback)

Step 6: Verify with simulation