Glossary

9.2 Active filter topologies and design

3 min read•august 9, 2024

Active filters are the superheroes of signal processing. They use op-amps to boost performance, allowing for steeper roll-offs and higher Q factors than passive filters. Plus, they're easier to tweak and don't load down your circuit.

This section dives into different active filter designs like Sallen-Key and multiple feedback. You'll learn how to choose the right topology for your needs, whether you're building a for audio or a for wireless comms.

Filter Types

Low-Pass and High-Pass Filters

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  • Low-pass filter allows frequencies below to pass through
    • Attenuates higher frequencies
    • Applications include audio systems, removing high-frequency noise
    • : H(s)=ωcs+ωcH(s) = \frac{\omega_c}{s + \omega_c}
  • permits frequencies above cutoff frequency to pass
    • Blocks lower frequencies
    • Used in audio crossovers, AC coupling circuits
    • Transfer function: H(s)=ss+ωcH(s) = \frac{s}{s + \omega_c}
  • Cutoff frequency (fcf_c) determines filter behavior
    • Calculated using and values: fc=12πRCf_c = \frac{1}{2\pi RC}
  • response varies with frequency
    • Low-pass: -20 dB/decade slope above cutoff
    • High-pass: -20 dB/decade slope below cutoff

Band-Pass and Band-Stop Filters

  • Band-pass filter allows a specific range of frequencies to pass
    • Combines low-pass and high-pass characteristics
    • Used in wireless communications, audio equalization
    • Transfer function: H(s)=Ass2+(ω0/Q)s+ω02H(s) = \frac{As}{s^2 + (\omega_0/Q)s + \omega_0^2}
  • Band-stop filter () attenuates a specific frequency range
    • Blocks unwanted frequencies while allowing others to pass
    • Applications include noise reduction, eliminating power line interference
    • Transfer function: H(s)=s2+ω02s2+(ω0/Q)s+ω02H(s) = \frac{s^2 + \omega_0^2}{s^2 + (\omega_0/Q)s + \omega_0^2}
  • Center frequency (f0f_0) and define filter response
    • Center frequency: f0=12πLCf_0 = \frac{1}{2\pi \sqrt{LC}}
    • Bandwidth: BW=f0QBW = \frac{f_0}{Q}
  • influences filter selectivity
    • Higher Q results in narrower bandwidth and steeper roll-off

Filter Characteristics

Filter Response Types

  • Butterworth filter provides maximally flat passband response
    • Smooth roll-off in stopband
    • Moderate selectivity and phase response
    • Transfer function: ∣H(jω)∣2=11+(ω/ωc)2n|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2n}}
  • Chebyshev filter offers steeper roll-off but with passband ripple
    • Type I: Ripple in passband, flat stopband
    • Type II: Flat passband, ripple in stopband
    • Improved selectivity compared to Butterworth
    • Transfer function: ∣H(jω)∣2=11+ϵ2Tn2(ω/ωc)|H(j\omega)|^2 = \frac{1}{1 + \epsilon^2 T_n^2(\omega/\omega_c)}

Filter Design Parameters

  • Filter order determines roll-off rate and stopband attenuation
    • Higher order increases slope steepness (n * -20 dB/decade)
    • Affects component count and circuit complexity
  • Cutoff frequency marks -3 dB point in frequency response
    • Defines boundary between passband and stopband
    • Calculated using component values in active filter circuits
  • Q factor measures filter's selectivity and bandwidth
    • Higher Q results in sharper resonance peak
    • Influences filter's transient response and ringing
    • Calculated as: Q=f0BWQ = \frac{f_0}{BW}

Active Filter Topologies

Sallen-Key and Multiple Feedback Topologies

  • offers simple design and low component count
    • Uses for improved performance
    • Can implement low-pass, high-pass, and band-pass filters
    • Transfer function (low-pass): H(s)=Kω02s2+2ζω0s+ω02H(s) = \frac{K\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}
  • Multiple feedback topology provides high Q factors and gain
    • Uses for stability
    • Suitable for band-pass and notch filter designs
    • Transfer function (band-pass): H(s)=−sR2C11+s(C1R1+C1R2+C2R2)+s2R1R2C1C2H(s) = \frac{-sR_2C_1}{1 + s(C_1R_1 + C_1R_2 + C_2R_2) + s^2R_1R_2C_1C_2}
  • Both topologies utilize operational amplifiers as active elements
    • Op-amps provide gain and impedance buffering
    • Enable higher Q factors compared to passive filters

Advanced Filter Structures

  • State variable filter offers simultaneous low-pass, high-pass, and band-pass outputs
    • Uses multiple op-amps for increased flexibility
    • Allows independent control of Q and cutoff frequency
    • Transfer function (band-pass): H(s)=Kω0ss2+ω0s/Q+ω02H(s) = \frac{K\omega_0s}{s^2 + \omega_0s/Q + \omega_0^2}
  • Biquad filter implements second-order transfer functions
    • Cascadable for higher-order filters
    • Provides low sensitivity to component variations
    • Transfer function: H(s)=b2s2+b1s+b0s2+a1s+a0H(s) = \frac{b_2s^2 + b_1s + b_0}{s^2 + a_1s + a_0}
  • Both structures offer improved performance and versatility
    • Used in , instrumentation, and communications systems
    • Allow realization of complex filter responses

Key Terms to Review (22)

Audio processing: Audio processing refers to the manipulation and transformation of audio signals to enhance or modify sound quality and characteristics. This process can involve filtering, equalization, dynamic range compression, and effects such as reverb or echo, which are essential in various applications from music production to telecommunications.
Band-pass filter: A band-pass filter is an electronic circuit that allows signals within a specific frequency range to pass through while attenuating frequencies outside that range. This type of filter is crucial in applications where you want to isolate a certain frequency band, such as in audio processing, communications, and signal processing. The design and component selection, as well as the filter topology, play significant roles in achieving the desired filtering characteristics.
Bandwidth: Bandwidth refers to the range of frequencies over which a system can operate effectively, often defined as the difference between the upper and lower frequency limits. It plays a crucial role in determining how a system responds to signals, influencing aspects like quality and performance across various applications.
Capacitor: A capacitor is a passive electronic component that stores electrical energy in an electric field, created by a pair of conductive plates separated by an insulating material known as a dielectric. Capacitors play a crucial role in various electrical and electronic applications, influencing behaviors such as energy storage, filtering, and timing within circuits.
Component Tolerances: Component tolerances refer to the permissible variations in the values of electronic components, such as resistors, capacitors, and inductors. These tolerances are critical in circuit design and can significantly affect the performance of active filters, as they define how much a component's actual value can deviate from its specified value. Understanding these tolerances is essential for ensuring that an active filter functions within its desired specifications.
Cutoff Frequency: Cutoff frequency is the frequency at which the output power of a filter or system drops to half its maximum value, typically corresponding to a -3 dB point in the magnitude response. It serves as a crucial parameter in determining how well a filter can pass or attenuate signals, linking it to key concepts like bandwidth, quality factor, and system response characteristics.
Frequency response equations: Frequency response equations describe how a system's output responds to various input frequencies. These equations help determine the behavior of circuits, particularly in the context of active filters, by analyzing their gain and phase shift across a range of frequencies, allowing designers to understand how well a circuit will perform in filtering signals.
Gain: Gain refers to the ratio of output signal power to input signal power in a circuit, indicating how much a system amplifies a signal. It is a crucial concept in understanding how circuits process signals, especially in applications involving filters, operational amplifiers, and analog signal processing. The gain can be expressed in linear terms or in decibels (dB), and it plays a vital role in determining the performance and characteristics of various electronic systems.
High-pass filter: A high-pass filter is an electronic circuit that allows signals with a frequency higher than a certain cutoff frequency to pass through while attenuating signals with lower frequencies. Understanding high-pass filters is crucial for analyzing magnitude and phase responses, designing effective circuits, and selecting the right components for specific applications.
Low-pass filter: A low-pass filter is an electronic circuit that allows signals with a frequency lower than a certain cutoff frequency to pass through while attenuating signals with frequencies higher than that threshold. This filtering process is crucial for various applications, including audio processing, signal conditioning, and noise reduction, helping to shape the frequency response of a system.
Negative feedback: Negative feedback is a process in control systems where the output of a system is fed back in a way that reduces the overall output. This mechanism helps stabilize systems by automatically correcting deviations from a desired level. It plays a crucial role in various applications, enabling improved accuracy and stability in performance.
Notch filter: A notch filter is a specific type of band-stop filter designed to attenuate a narrow range of frequencies while allowing other frequencies to pass through unaffected. This makes it particularly useful for eliminating unwanted signals, such as noise or interference, without disrupting the overall frequency response of a system. Notch filters are often implemented in various applications including audio processing, telecommunications, and instrumentation.
Operational Amplifier: An operational amplifier, often abbreviated as op-amp, is a high-gain voltage amplifier with differential inputs and usually a single-ended output. They are crucial components in analog electronics, allowing for the implementation of various signal processing functions like amplification, filtering, and mathematical operations such as addition and integration. Their versatility makes them integral in both passive and active filter designs.
Phase Shift: Phase shift refers to the amount by which a waveform is shifted horizontally from a reference point, typically measured in degrees or radians. In the context of electrical circuits, phase shifts are critical for understanding how different components interact with alternating current (AC) signals, particularly when analyzing quality factors, resonance, filter design, and frequency responses.
Positive feedback: Positive feedback is a process where the output of a system amplifies its own input, leading to an increase in activity or effect within the system. This mechanism can lead to exponential growth or runaway effects, significantly influencing the behavior of electronic circuits and systems. In many cases, positive feedback can enhance performance, but it also risks instability if not controlled properly.
Power Supply Requirements: Power supply requirements refer to the necessary electrical specifications that must be met for a device or circuit to function correctly, including voltage, current, and power ratings. These requirements are crucial in the design and operation of active filter circuits, as they determine the stability, performance, and overall efficiency of the filter system. Understanding these specifications helps ensure that active filters perform as intended within electronic systems.
Q factor: The q factor, or quality factor, is a dimensionless parameter that describes the damping of an oscillator or resonator, defining its bandwidth relative to its center frequency. A higher q factor indicates a lower rate of energy loss relative to the stored energy, resulting in a narrower bandwidth and sharper resonance. This concept is critical in filter design, influencing how effectively filters can isolate or reject specific frequency ranges.
Resistor: A resistor is a passive electrical component that resists the flow of electric current, converting electrical energy into heat. It plays a vital role in controlling current and voltage levels in circuits, impacting how components work together. Resistors are essential for setting bias points in active devices, limiting current to protect components, and shaping signals within various electronic applications.
Sallen-key topology: Sallen-Key topology is an active filter design technique that employs operational amplifiers to create second-order low-pass and high-pass filters. It provides a simple and effective way to design filters with desired frequency characteristics using minimal components, thus offering flexibility in component selection and adjustment of filter parameters.
Signal conditioning: Signal conditioning refers to the manipulation of an analog signal to prepare it for further processing, measurement, or transmission. This process often involves amplifying, filtering, and converting signals to improve their accuracy and reliability, ensuring that the signals are suitable for subsequent stages in electronic systems. Signal conditioning is essential in applications where raw signals are affected by noise or require conversion to a different format.
State-variable filters: State-variable filters are a type of active filter that can implement various frequency responses (low-pass, high-pass, band-pass, and notch) using a common topology. They utilize state variables to describe the behavior of the filter, allowing for easy manipulation and adjustment of filter characteristics through feedback and gain settings. This flexibility makes them highly suitable for applications in signal processing and audio systems.
Transfer Function: A transfer function is a mathematical representation that defines the relationship between the input and output of a linear time-invariant (LTI) system in the frequency domain. It captures how a system responds to various frequencies, providing insights into system behavior, stability, and dynamics.
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