Active Filters

Introduction

Introduction to Active Filters

  • Filters process signals by passing selected frequencies while rejecting others

  • Active filters use op-amps with passive RC circuits

  • Filters categorized by output voltage variation with input frequency

  • Four basic categories:

    • Low-pass

    • High-pass

    • Band-pass

    • Band-stop

  • Advantages over passive filters:

    • Gain provided by active elements

    • High input impedance

    • Low output impedance

Filter Responses

Low-Pass Filter Response

  • Passes frequencies from DC to critical frequency \(f_c\), attenuates others

  • Passband: Frequencies with \(< -3 \, \text{dB}\) attenuation

  • Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\) (single-pole \(RC\) filter)

  • Roll-off: \(-20 \, \text{dB/decade}\) per pole

  • Bandwidth: \(BW = f_c\) (ideal)

Low-pass filter responses
Low-pass filter responses

High-Pass Filter Response

  • Passes frequencies above \(f_c\), attenuates below

  • Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\) (single-pole \(RC\) filter)

  • Roll-off: \(-20 \, \text{dB/decade}\) per pole

  • Practical limitation: Op-amp’s internal \(RC\) circuits limit high-frequency response

High-pass filter responses
High-pass filter responses

Band-Pass Filter Response

  • Passes frequencies between lower (\(f_{c1}\)) and upper (\(f_{c2}\)) critical frequencies

  • Bandwidth: \(BW = f_{c2} - f_{c1}\)

  • Center frequency: \(f_0 = \sqrt{f_{c1} f_{c2}}\)

  • Quality factor: \(Q = \dfrac{f_0}{BW} = \dfrac{1}{DF}\) (DF: Damping factor)

  • \(Q > 10\): Narrow-band; \(Q < 10\): Wide-band

Band-pass filter responses
Band-pass filter responses

Band-Stop Filter Response

  • Rejects frequencies between \(f_{c1}\) and \(f_{c2}\), passes others

  • Bandwidth: \(BW = f_{c2} - f_{c1}\)

  • Opposite of band-pass filter

Band-stop filter response
Band-stop filter response

Filter Characteristics

Filter Response Characteristics

  • Butterworth: Flat passband, \(-20 \, \text{dB/decade/pole}\), nonlinear phase

  • Chebyshev: Steeper roll-off (\(> -20 \, \text{dB/decade/pole}\)), passband ripples

  • Bessel: Linear phase, minimal overshoot, suitable for pulse waveforms

Comparative filter responses
Comparative filter responses

Damping Factor

  • Determines response characteristic

  • Defined as: \(DF = 2 - \dfrac{R_1}{R_2}\)

  • Example: For second-order Butterworth, \(DF = 1.414\), so \(\dfrac{R_1}{R_2} = 0.586\)

General active filter
General active filter

Critical Frequency and Filter Order

  • Critical Frequency (\(f_c\)) for first-order RC filter: \[f_c = \frac{1}{2\pi RC}\]

  • Applies to both low-pass and high-pass single-pole configurations

  • Roll-Off Rate:

    • Butterworth filter: \(-20\, \text{dB/decade/pole}\)

    • 1st-order (1 pole): \(-20\, \text{dB/decade}\)

    • 2nd-order (2 poles): \(-40\, \text{dB/decade}\)

    • 3rd-order (3 poles): \(-60\, \text{dB/decade}\), etc.

Filter Implementations

Single-Pole Low-Pass Filter

  • Uses one \(RC\) circuit, roll-off \(-20 \, \text{dB/decade}\)

  • Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\)

  • Gain: \(A_{cl(NI)} = \dfrac{R_1}{R_2} + 1\)

Single-pole active low-pass filter
Single-pole active low-pass filter

Sallen-Key Low-Pass Filter

  • Common second-order configuration (VCVS)

  • Two-pole filter, roll-off \(-40 \, \text{dB/decade}\)

  • Critical frequency: \(f_c = \dfrac{1}{2 \pi \sqrt{R_A R_B C_A C_B}}\)

  • If \(R_A = R_B = R\), \(C_A = C_B = C\): \(f_c = \dfrac{1}{2 \pi R C}\)

Sallen-Key low-pass filter
Sallen-Key low-pass filter

Cascaded Low-Pass Filters

  • Third-order: Cascade 2-pole Sallen-Key + 1-pole filter (\(-60 \, \text{dB/decade}\))

  • Fourth-order: Cascade two 2-pole Sallen-Key filters (\(-80 \, \text{dB/decade}\))

Higher-order low-pass filters
Higher-order low-pass filters

Single-Pole High-Pass Filter

  • Roll-off: \(-20 \, \text{dB/decade}\)

  • Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\)

  • Limitation: Op-amp’s internal \(RC\) limits high-frequency response

Single-pole high-pass filter
Single-pole high-pass filter

Sallen-Key High-Pass Filter

  • Two-pole filter, roll-off \(-40 \, \text{dB/decade}\)

  • \(R\) and \(C\) positions swapped compared to low-pass

  • Response optimized by feedback resistors \(R_1\), \(R_2\)

Sallen-Key high-pass filter
Sallen-Key high-pass filter

Band-Pass and Band-Stop Filters

Cascaded Band-Pass Filter

  • Cascade high-pass and low-pass Sallen-Key filters

  • \(f_{c1} = \dfrac{1}{2 \pi \sqrt{R_{A1} R_{B1} C_{A1} C_{B1}}}\)

  • \(f_{c2} = \dfrac{1}{2 \pi \sqrt{R_{A2} R_{B2} C_{A2} C_{B2}}}\)

  • Center frequency: \(f_0 = \sqrt{f_{c1} f_{c2}}\)

  • Suitable for wide bandwidth applications

Cascaded band-pass filter
Cascaded band-pass filter

Multiple-Feedback Band-Pass Filter

  • Center frequency: \(f_0 = \dfrac{1}{2 \pi C \sqrt{\dfrac{R_1 + R_3}{R_1 R_2 R_3}}}\)

  • Gain: \(A_0 = \dfrac{R_2}{2 R_1}\)

  • Resistors:

    • \(R_1 = \dfrac{Q}{2 \pi f_0 C A_0}\)

    • \(R_2 = \dfrac{Q}{\pi f_0 C}\)

    • \(R_3 = \dfrac{Q}{2 \pi f_0 C (2 Q^2 - A_0)}\)

Multiple-feedback band-pass
Multiple-feedback band-pass

State-Variable Filter

  • Uses summing amplifier and two integrators

  • Center frequency: \(f_0 \approx \dfrac{1}{2 \pi R C}\)

  • \(Q = \dfrac{1}{3} \left( \dfrac{R_5}{R_6} + 1 \right)\)

  • Provides low-pass, high-pass, band-pass outputs

State-variable filter
State-variable filter

Biquad Filter

  • Integrator, inverting amplifier, integrator

  • Provides band-pass and low-pass outputs

  • High \(Q\), bandwidth independent of \(f_0\)

Biquad filter
Biquad filter

Multiple-Feedback Band-Stop Filter

  • Similar to band-pass, with modified resistor configuration

Multiple-feedback band-stop
Multiple-feedback band-stop

State-Variable Band-Stop Filter

  • Created by summing LP and HP outputs

  • Common application: 60 Hz noise rejection

  • Center frequency set by integrator RC circuits

State-variable band-stop
State-variable band-stop

Measurements

Filter Response Measurements

Discrete Point Measurement

  • Set constant input amplitude

  • Sweep frequency in steps

  • Record output at each frequency

  • Plot response curve

Discrete measurement
Discrete measurement

Swept Frequency Measurement

  • More efficient than discrete method

  • Uses swept frequency generator

  • Display options:

    • Spectrum analyzer

    • Oscilloscope in X-Y mode

  • Provides continuous response curve

Swept frequency measurement
Swept frequency measurement

Summary

  • Covered low-pass, high-pass, band-pass, band-stop filters

  • Discussed Butterworth, Chebyshev, Bessel characteristics

  • Analyzed single-pole, Sallen-Key, multiple-feedback, state-variable, biquad filters

  • Explained damping factor, critical frequency, roll-off rates

  • Described discrete point and swept frequency measurements