Electronics Guide

Active filters combine passive components with active devices, typically operational amplifiers, to achieve frequency-selective behavior with significant advantages over purely passive implementations. By incorporating gain elements, active filters can provide signal amplification while filtering, eliminate the need for bulky and lossy inductors at low frequencies, achieve precise frequency response characteristics, and offer easy tunability. These attributes make active filters the preferred choice for most audio, instrumentation, and low-frequency signal processing applications.

The design of active filters involves selecting an appropriate topology, determining the filter order and response type, and calculating component values to achieve the desired cutoff frequency, quality factor (Q), and gain. This article explores the major active filter topologies, their characteristics, design equations, and practical implementation considerations that enable engineers to create precise frequency-selective circuits for diverse applications.

Fundamentals of Active Filter Design

Active filters use operational amplifiers or other active devices in conjunction with resistors and capacitors to implement filter transfer functions. The active device provides gain that compensates for losses in the RC network and enables the realization of complex pole locations that would require inductors in a passive implementation.

Key Parameters

Several parameters characterize active filter performance:

• Cutoff Frequency (fc): The frequency at which the filter response is 3 dB below the passband level, marking the transition between passband and stopband
• Quality Factor (Q): A measure of the sharpness of the filter's resonance peak, with higher Q values producing steeper transitions and more pronounced peaks
• Passband Gain: The amplification provided by the filter in the passband region
• Filter Order: The number of poles in the transfer function, determining the ultimate rolloff rate (20 dB/decade per pole)
• Sensitivity: How much the filter response changes with component value variations

Biquadratic Sections

Most active filter designs are based on second-order sections, also called biquadratic or biquad sections. The general second-order transfer function takes the form:

H(s) = (a2s^2 + a1s + a0) / (s^2 + (w0/Q)s + w0^2)

where w0 is the natural frequency and Q is the quality factor. By choosing appropriate values for the numerator coefficients (a0, a1, a2), different filter types can be realized: low-pass, high-pass, band-pass, band-stop (notch), and all-pass. Higher-order filters are constructed by cascading multiple second-order sections, with each section contributing a pair of poles to the overall response.

Sallen-Key Configuration

The Sallen-Key topology, developed by R.P. Sallen and E.L. Key at MIT Lincoln Laboratory in 1955, is one of the most popular active filter configurations. It uses a single operational amplifier in a voltage follower or non-inverting amplifier configuration, with an RC network providing the frequency-selective behavior. The Sallen-Key structure is valued for its simplicity, low component count, and ease of design.

Low-Pass Sallen-Key Filter

The low-pass Sallen-Key filter consists of two resistors and two capacitors arranged around a non-inverting amplifier. The basic configuration places one resistor-capacitor pair in series from input to the non-inverting input, with a second capacitor from this node to ground, and another resistor providing positive feedback from the output. The transfer function provides a second-order low-pass response with:

• Cutoff frequency: fc = 1 / (2 * pi * sqrt(R1 * R2 * C1 * C2))
• Quality factor: Q = sqrt(R1 * R2 * C1 * C2) / (C2 * (R1 + R2) + R1 * C1 * (1 - K))
• DC gain: K = 1 + Rf / Rg (for non-inverting configuration)

For the unity-gain configuration (voltage follower), K = 1, and the quality factor depends only on the passive component ratios. This simplifies tuning but limits Q to relatively low values. For higher Q values, gain is increased by adding feedback resistors, though stability margins must be carefully managed.

Equal-Value Component Design

A practical design approach uses equal-value resistors (R1 = R2 = R) and equal-value capacitors (C1 = C2 = C), which simplifies component procurement and matching. With this constraint, the cutoff frequency becomes fc = 1 / (2 * pi * R * C), and the quality factor for the unity-gain version is fixed at Q = 0.5, corresponding to a critically damped response. To achieve Q values required for Butterworth (Q = 0.707) or Chebyshev responses, gain must be added.

High-Pass Sallen-Key Filter

The high-pass version exchanges the positions of resistors and capacitors in the low-pass topology. Capacitors now appear in series with the signal path, blocking DC and low frequencies while passing high frequencies. The design equations are analogous to the low-pass case, with the roles of R and C interchanged in the component ratio expressions.

Sensitivity Analysis

Sallen-Key filters exhibit moderate sensitivity to component variations. The cutoff frequency sensitivity is typically 0.5 (a 1% change in component value causes approximately 0.5% change in cutoff frequency). Q sensitivity is higher, particularly at high Q values, making the Sallen-Key topology better suited for applications with Q less than 10. For higher Q requirements, state-variable or biquad topologies offer better performance.

Multiple Feedback Configuration

The multiple feedback (MFB) topology, also known as the infinite-gain multiple feedback or Rauch filter, uses an operational amplifier in an inverting configuration with multiple feedback paths. This arrangement provides both the desired filter response and signal inversion, making it useful in applications where phase inversion is acceptable or desired.

Low-Pass MFB Filter

The low-pass MFB filter uses three resistors and two capacitors. One capacitor provides feedback from output to the inverting input, while the second capacitor connects from the junction of two input resistors to ground. The third resistor provides additional feedback. This configuration yields a second-order low-pass response with:

• Cutoff frequency: fc = 1 / (2 * pi * sqrt(R2 * R3 * C1 * C2))
• Quality factor: Q = sqrt(R2 * R3 * C1 * C2) / (C1 * (R2 + R3) + R3 * C2 * K)
• DC gain: K = -R3 / R1 (inverting)

Band-Pass MFB Filter

The MFB topology readily implements band-pass filters by placing a capacitor in series with the input resistor. This creates a response that peaks at the center frequency and rolls off on both sides. The band-pass MFB filter is widely used in tone detection circuits, audio equalizers, and communication systems.

Advantages and Limitations

The MFB topology offers several advantages:

• Lower sensitivity: Generally exhibits lower sensitivity to component variations than Sallen-Key, especially for Q values above 2
• Good high-frequency performance: The inverting configuration provides better bandwidth utilization of the op-amp
• Inherent gain: Provides adjustable gain as part of the filter function

Limitations include the inverting output (180-degree phase shift), somewhat more complex design equations, and the need for a virtual ground at the op-amp input that may require careful impedance management.

State-Variable Filters

State-variable filters use multiple operational amplifiers configured as integrators and summers to implement second-order filter functions. This topology derives its name from the state-space representation of differential equations, where each integrator output represents a state variable. State-variable filters provide simultaneous low-pass, band-pass, and high-pass outputs from a single circuit, offering exceptional flexibility and tunability.

Basic State-Variable Structure

The classic state-variable filter uses three operational amplifiers:

• First amplifier: Configured as a summing amplifier that combines the input signal with feedback from the filter outputs
• Second amplifier: Configured as an integrator, producing the band-pass output
• Third amplifier: Also an integrator, producing the low-pass output

The high-pass output is available at the summing amplifier output. Each output provides a specific frequency response characteristic, all sharing the same cutoff frequency and Q value.

Frequency and Q Control

The cutoff frequency is set by the integrator time constants: fc = 1 / (2 * pi * R * C) for each integrator. Changing both integrator resistors simultaneously tunes the frequency without affecting Q. The quality factor is determined by the feedback network around the summing amplifier, typically: Q = (R4 + R5) / (R4 * (1 + R5/R6)), where R4, R5, and R6 are the summing network resistors.

This separation of frequency and Q controls is a major advantage of the state-variable topology, enabling independent adjustment of these parameters. Some implementations use potentiometers for both controls, facilitating easy tuning in laboratory and prototyping environments.

Notch and All-Pass Outputs

By summing the low-pass and high-pass outputs, a band-stop (notch) response is obtained. The notch frequency equals the filter's center frequency, with the notch depth and width determined by the summing amplifier gain and the filter Q. Similarly, combining all three outputs with appropriate scaling can produce an all-pass response that maintains constant amplitude while introducing frequency-dependent phase shift.

Biquad Filters

Biquad filters (biquadratic filters) represent a general class of second-order active filters that implement the standard biquadratic transfer function. While the term "biquad" technically applies to any second-order filter, it commonly refers to specific topologies that provide independent control of all transfer function coefficients.

Standard Biquad Configuration

The standard biquad uses two integrators in a feedback loop, similar to the state-variable filter, but with modified input and feedback arrangements. The circuit typically provides access to adjust poles and zeros independently, enabling implementation of arbitrary second-order transfer functions including low-pass, high-pass, band-pass, notch, and all-pass responses.

Programming Biquad Parameters

Biquad filters are particularly useful when the filter response must be precisely controlled or changed during operation. The key parameters are:

• Center frequency (f0): Controlled by integrator resistors or capacitors
• Quality factor (Q): Set by feedback resistor ratios
• Zero locations: Determined by input summing network coefficients
• Gain: Adjusted independently of frequency response shape

This parameter independence makes biquads ideal for adaptive filters, tunable equalizers, and applications where filter characteristics must be adjusted in real-time.

Tow-Thomas and KHN Structures

The Tow-Thomas and Kerwin-Huelsman-Newcomb (KHN) filters represent refined state-variable implementations that offer specific advantages in practical applications.

Tow-Thomas Biquad

The Tow-Thomas biquad, developed independently by Josef Tow and L.C. Thomas in 1968, uses two integrators and a gain stage to implement a second-order transfer function. The circuit provides low-pass and band-pass outputs simultaneously, with the following characteristics:

• Independent frequency tuning: A single resistor change adjusts cutoff frequency without affecting Q
• Independent Q adjustment: A separate resistor controls Q without affecting frequency
• Low sensitivity: Exhibits low sensitivity to component tolerances
• Wide tuning range: Can achieve frequency ranges exceeding 1000:1 with proper component selection

The Tow-Thomas topology is widely used in voltage-controlled filters for synthesizers and in tunable electronic crossover networks. The frequency control resistor can be replaced with a voltage-controlled element such as an operational transconductance amplifier (OTA) for electronic tuning.

KHN Filter

The Kerwin-Huelsman-Newcomb (KHN) filter, published in 1967, provides the complete set of second-order outputs (low-pass, band-pass, and high-pass) with excellent parameter control. The standard KHN configuration uses three operational amplifiers and offers:

• Simultaneous outputs: All three basic filter types available concurrently
• Orthogonal tuning: Frequency and Q adjustable without interaction
• Low output impedance: Each output comes directly from an op-amp
• Easy notch creation: Summing LP and HP outputs produces notch response

The KHN structure forms the basis for many commercial universal active filter integrated circuits, which include on-chip resistor networks for convenient frequency and Q programming.

All-Pass and Phase-Shift Networks

All-pass filters pass all frequencies with equal amplitude while introducing frequency-dependent phase shift. Unlike conventional filters that modify amplitude response, all-pass networks are used specifically to control signal phase without affecting magnitude. Applications include group delay equalization, phase compensation in feedback systems, and creating specific phase relationships in audio and communication systems.

First-Order All-Pass Filter

A first-order all-pass filter provides 180 degrees of phase shift across its operating range, transitioning from 0 degrees at DC to 180 degrees at high frequencies, with 90 degrees shift occurring at the characteristic frequency. The basic implementation uses a single op-amp with an RC network:

• Transfer function: H(s) = (s - w0) / (s + w0)
• Magnitude: |H(jw)| = 1 for all frequencies
• Phase: Varies continuously from 0 to -180 degrees
• Characteristic frequency: f0 = 1 / (2 * pi * R * C)

Second-Order All-Pass Filter

Second-order all-pass filters provide 360 degrees of phase shift range and can be designed with specific group delay characteristics. The transfer function includes a quadratic numerator and denominator with matched coefficients but opposite signs for the linear terms. Implementation typically requires two operational amplifiers and provides controllable phase transition rate through the Q parameter.

Group Delay Equalization

Group delay is the derivative of phase with respect to frequency, representing the time delay experienced by signal components at different frequencies. Non-constant group delay causes signal distortion in wideband systems. All-pass filters can be designed to add complementary group delay, flattening the overall system response. This technique is essential in digital communication systems, high-fidelity audio, and radar signal processing.

Phase-Shift Oscillators

Cascading multiple first-order all-pass sections creates a phase-shift network that can form the basis of an oscillator. When the total phase shift reaches 360 degrees at a specific frequency, and the loop gain equals unity, the system oscillates. This principle underlies simple phase-shift oscillators used in audio and low-frequency test applications.

Switched-Capacitor Filters

Switched-capacitor filters replace resistors with capacitors that are periodically switched by clock signals, using the principle that a capacitor switched at high frequency between two nodes behaves like a resistor. This technique enables the integration of precision analog filters on standard digital CMOS processes without requiring the accurate resistors that are difficult to fabricate in integrated circuits.

Operating Principle

When a capacitor C is switched at frequency fclk between two voltage nodes, it transfers charge at an average rate equivalent to a resistor of value:

Req = 1 / (fclk * C)

The filter's cutoff frequency becomes proportional to the clock frequency rather than absolute RC time constants. This ratio-metric behavior means that filter characteristics depend only on capacitor ratios (which can be controlled precisely in IC fabrication) and the clock frequency (which can be generated accurately). This principle enables:

• Accurate filter responses: Better than 1% accuracy achievable without trimming
• Clock-tunable filters: Cutoff frequency tracks clock frequency precisely
• Monolithic integration: Entire filters implemented on single IC
• Low power operation: CMOS switches consume minimal power

Anti-Aliasing Requirements

Because switched-capacitor filters sample the input signal at the clock frequency, they are subject to aliasing effects. The clock frequency must be significantly higher than the highest signal frequency (typically 50 to 100 times the filter cutoff frequency) to prevent aliasing artifacts. Many applications require a continuous-time anti-aliasing filter ahead of the switched-capacitor filter to attenuate high-frequency components that could fold back into the passband.

Clock Feedthrough

Switching transients can couple through to the output, creating artifacts at the clock frequency and its harmonics. Careful circuit design, balanced switch configurations, and post-filtering minimize this effect. Modern switched-capacitor filter ICs incorporate internal measures to reduce clock feedthrough to acceptable levels for most applications.

Common Applications

Switched-capacitor filters are extensively used in:

• Anti-aliasing for ADCs: Integrated filters preceding analog-to-digital converters
• Voice-band filtering: Telephone and audio codec applications
• Data acquisition systems: Programmable signal conditioning
• Communication receivers: Channel selection and IF filtering

Commercial switched-capacitor filter ICs are available with configurable response types, adjustable cutoff frequencies, and multiple filter orders, providing convenient solutions for many filtering requirements.

Tunable and Programmable Filters

Many applications require filters whose characteristics can be adjusted during operation, either manually for calibration and setup or automatically as part of adaptive signal processing. Several techniques enable real-time filter tuning while maintaining stable and predictable performance.

Voltage-Controlled Filters

Voltage-controlled filters (VCFs) use voltage-variable elements to adjust filter parameters in response to control signals. Common approaches include:

• OTA-based filters: Operational transconductance amplifiers whose transconductance varies with a control current, enabling voltage control of filter cutoff frequency
• JFET variable resistors: Junction field-effect transistors operating in their ohmic region as voltage-controlled resistors
• Multiplying DAC tuning: Digital-to-analog converters used to scale reference voltages and effectively vary component values
• Switched-capacitor clock tuning: Varying the clock frequency of switched-capacitor filters to shift cutoff frequency

Digitally Programmable Filters

Digitally programmable filters use digital control signals to select component values or filter configurations. Implementation methods include:

• Switched resistor/capacitor arrays: Banks of components selected by analog switches under digital control
• Digital potentiometers: Programmable resistive dividers that replace fixed resistors
• DAC-controlled VCFs: Digital-to-analog converters generating control voltages for voltage-controlled filters
• Filter coefficient registers: Dedicated filter ICs with digitally loaded coefficients

Automatic Tuning Systems

Some filter systems incorporate automatic tuning to compensate for component variations, temperature drift, or aging effects. A reference signal at a known frequency is applied to the filter, and feedback circuits adjust component values to maintain the desired response. This technique is commonly used in high-performance communication receivers and precision measurement systems.

Gyrator-Based Inductance Simulation

At low frequencies, inductors become impractically large, lossy, and expensive. Gyrators are active circuits that simulate inductive behavior using capacitors, enabling the implementation of inductor-based filter topologies without actual inductors. This technique is essential for realizing LC filter responses at audio and sub-audio frequencies.

Gyrator Fundamentals

A gyrator is a two-port network that transforms impedance by inversion. When a capacitor C is connected to one port of an ideal gyrator with gyration resistance R, the impedance seen at the other port is:

Zsim = R^2 * s * C = s * Lsim

This represents an inductor with equivalent inductance Lsim = R^2 * C. By choosing appropriate values of R and C, inductances from millihenries to thousands of henries can be simulated, values that would be impractical with physical inductors.

Single Op-Amp Gyrator

The simplest gyrator implementation uses a single operational amplifier with two resistors and a capacitor. The circuit presents an inductive impedance at its input that can replace an inductor in many filter applications. Limitations include:

• Grounded inductance only: One terminal must connect to ground
• Finite Q: Real op-amp limitations cause losses equivalent to a series resistance
• Limited frequency range: Op-amp bandwidth constrains upper frequency limit

Floating Inductance Simulation

Applications requiring inductors with neither terminal grounded (floating inductors) need more complex circuits. Two-op-amp and four-op-amp configurations can simulate floating inductors, enabling implementation of bridged-T, twin-T, and other topologies that require non-grounded reactive elements.

Applications

Gyrator-simulated inductors are commonly used in:

• Audio equalizers: Parametric and graphic equalizer sections
• Telephone hybrid circuits: Impedance matching networks
• Active crossover networks: Speaker system frequency dividers
• Low-frequency oscillators: LC oscillator circuits below audio frequencies

Generalized Impedance Converters

Generalized impedance converters (GICs) extend the gyrator concept to provide more flexible impedance transformation capabilities. A GIC can multiply or divide an impedance by a factor that may itself be frequency-dependent, enabling creation of frequency-dependent negative resistors (FDNRs) and other exotic elements useful in filter synthesis.

GIC Structure

The standard GIC uses two operational amplifiers and five impedances (resistors and capacitors) arranged in a specific topology. The input impedance looking into one terminal depends on the product and quotient of the five impedances:

Zin = (Z1 * Z3 * Z5) / (Z2 * Z4)

By selecting which elements are resistors and which are capacitors, various impedance behaviors can be created:

• Inductance simulation: Making Z4 a capacitor and others resistors yields inductive input impedance
• FDNR (D element): Making Z2 and Z4 capacitors creates an element with impedance proportional to 1/s^2
• Super-capacitor: Other combinations yield capacitive behavior with enhanced or unusual characteristics

FDNR Filter Design

The frequency-dependent negative resistor (FDNR, or D element) has impedance Z = D/s^2, behavior that does not exist in passive components. FDNR-based filter design applies a transformation to passive LC filter prototypes:

• Inductors in the prototype become resistors in the active realization
• Capacitors become FDNR elements (implemented with GICs)
• Resistors become capacitors

This transformation produces filters with characteristics identical to the passive prototype but using only resistors, capacitors, and op-amps. The approach is particularly effective for implementing high-order Butterworth, Chebyshev, and elliptic filters with accurate response shapes.

Practical Considerations

GIC-based filters offer excellent performance but require careful attention to:

• Component matching: Accuracy of the impedance transformation depends on precise component ratios
• Op-amp selection: Both amplifiers must have adequate bandwidth and output drive capability
• DC bias: Proper biasing ensures op-amps operate in their linear range
• Noise: Multiple amplifier stages can accumulate noise, requiring low-noise designs for sensitive applications

Practical Implementation Considerations

Translating active filter designs into working circuits requires attention to several practical factors that affect real-world performance.

Op-Amp Selection

The operational amplifier choice significantly impacts filter performance:

• Gain-bandwidth product: Should be at least 50-100 times the filter cutoff frequency to minimize response deviations
• Slew rate: Must accommodate the maximum signal rate of change to prevent distortion
• Input offset voltage: Critical for DC-accurate low-frequency filters
• Input bias current: Affects high-impedance filter sections; consider FET-input devices
• Noise: Both voltage and current noise contribute to output noise

Component Selection

Passive component quality affects filter accuracy and stability:

• Resistor tolerance: 1% metal film resistors suitable for most applications; 0.1% for precision filters
• Capacitor type: NPO/C0G ceramic or polypropylene film for stability; avoid Z5U and Y5V ceramics in frequency-determining positions
• Temperature coefficients: Match coefficients between components to minimize drift over temperature
• Capacitor ESR: Significant at high frequencies; affects Q and response accuracy

Layout Considerations

PCB layout affects filter performance, particularly at higher frequencies:

• Ground planes: Provide low-impedance return paths and shielding
• Component placement: Keep frequency-determining components close together to minimize parasitic effects
• Power supply decoupling: Local bypass capacitors for each op-amp prevent supply-coupled oscillation
• Input/output separation: Prevent coupling between high-gain sections

Testing and Tuning

Verifying filter performance requires appropriate measurement techniques:

• Frequency response: Network analyzers or swept-frequency generators with level detectors
• Phase response: Vector network analyzers or oscilloscope phase measurement
• Transient response: Step and pulse testing reveals ringing and settling behavior
• Noise measurement: Spectrum analyzers or specialized noise measurement systems

Comparing Active Filter Topologies

Different topologies offer distinct advantages for specific applications:

The choice depends on required performance, available component count and power budget, output phase requirements, tunability needs, and integration constraints.

Conclusion

Active filter topologies provide powerful tools for implementing precise frequency-selective circuits across a broad range of applications. From the simple Sallen-Key configuration suitable for basic filtering tasks to sophisticated state-variable and GIC-based structures for demanding applications, engineers have numerous options for realizing desired frequency responses with the advantages of gain, low-frequency operation without inductors, and ease of tunability.

Successful active filter design requires understanding both the theoretical foundations, including transfer functions, pole-zero placement, and sensitivity analysis, and practical considerations such as op-amp selection, component tolerances, and layout techniques. Modern design tools, including filter design software and simulation packages, simplify the process of selecting topologies, calculating component values, and verifying performance before committing to hardware.

As electronic systems continue to demand more precise signal conditioning and filtering, active filter technology remains essential. Whether implementing anti-aliasing filters for data converters, audio equalizers for sound systems, or channel filters for communication receivers, the topologies presented here provide the foundation for meeting diverse filtering requirements with accuracy, flexibility, and cost-effectiveness.

Related Topics

• Filter Design and Implementation
• Passive Filter Networks
• Operational Amplifiers and Linear Circuits
• Analog Electronics