Designing a Second-Order Butterworth Active High-Pass Filter

Comprehensive Guide to Achieving a 5 kHz Cut-Off Frequency

Key Takeaways

Butterworth Filter Characteristics: Ensures a maximally flat frequency response in the passband with a roll-off rate of 40 dB/decade.
Sallen-Key Topology: Preferred for its simplicity and stability in implementing second-order active filters.
Component Selection: Accurate resistor and capacitor values are crucial for achieving the desired cut-off frequency and filter performance.

Introduction to Butterworth High-Pass Filters

Butterworth filters are renowned for their maximally flat frequency response in the passband, making them ideal for applications requiring a smooth passband without ripples. A second-order Butterworth active high-pass filter not only attenuates frequencies below its cut-off point but also allows higher frequencies to pass with minimal loss. Designing such a filter for a specific cut-off frequency, like 5 kHz, involves careful selection of components and topology to ensure optimal performance.

Understanding the Filter Specifications

Key Parameters

Cut-Off Frequency (f_c): 5 kHz
Filter Order: Second-order, which provides a roll-off rate of 40 dB/decade or 12 dB/octave.
Tolerance: Component tolerances affect the precision of the cut-off frequency and overall filter performance.
Quality Factor (Q): For a Butterworth filter, Q = 1/√2 ≈ 0.707.

Filter Topology Selection

Why Choose Sallen-Key Configuration?

The Sallen-Key topology is widely favored for its simplicity, stability, and ease of implementation in active filter designs. It utilizes an operational amplifier (op-amp) along with resistors and capacitors to achieve the desired filtering characteristics. This configuration is particularly suitable for second-order filters, providing the necessary roll-off rate and phase response required for high-pass filtering applications.

Deriving the Transfer Function

Mathematical Foundation

The transfer function of a second-order Butterworth high-pass filter is given by:

$$ H(s) = \frac{s^2}{s^2 + \sqrt{2} \omega_c s + \omega_c^2} $$

Where:

ω_c: Angular cut-off frequency, ω_c = 2πf_c
Q: Quality factor, Q = 1/√2 ≈ 0.707 for Butterworth response

Component Selection and Calculation

Determining Resistor and Capacitor Values

The cut-off frequency for a Sallen-Key high-pass filter is determined by the resistor and capacitor values. The standard formula for the cut-off frequency is:

$$ f_c = \frac{1}{2\pi \sqrt{R_1 R_2 C_1 C_2}} $$

Assuming equal resistor (R₁ = R₂ = R) and capacitor (C₁ = C₂ = C) values simplifies the equation to:

$$ f_c = \frac{1}{2\pi R C} $$

Rearranging to solve for R and C:

$$ R C = \frac{1}{2\pi f_c} $$

Substituting f_c = 5 kHz:

$$ R C = \frac{1}{2\pi \times 5000} \approx 3.183 \times 10^{-5}\ \text{s} $$

Selecting a standard capacitor value, for example, C = 10 nF (10 × 10^-9 F), we can calculate R:

$$ R = \frac{3.183 \times 10^{-5}}{10 \times 10^{-9}} = 3.183\ \text{k}\Omega $$

Rounding to the nearest standard resistor value, we choose R = 3.2 kΩ.

Selecting the Operational Amplifier

Criteria for Op-Amp Selection

Gain-Bandwidth Product (GBW): Should be significantly higher than the cut-off frequency to ensure accurate amplification without phase shifts.
Low Noise: Minimizes the introduction of unwanted noise into the filter.
Supply Voltage: Compatible with the intended application’s power supply.
Stability: Ensures the filter operates without oscillations or distortions.

Common op-amp choices like the TL072 or LM741 are suitable due to their appropriate GBW and reliability.

Assembling the Filter Circuit

Step-by-Step Assembly Guide

Connect the Capacitors: Place C₁ and C₂ in series with the input signal.
Attach the Resistors: Connect R₁ and R₂ in parallel with C₁ and C₂, respectively.
Op-Amp Configuration: Connect the non-inverting input (+) of the op-amp to the junction of R₁ and R₂.
Feedback Loop: Connect the output of the op-amp to the inverting input (-) through a feedback resistor. For unity gain, this resistor can be equal to R₁.
Grounding: Ground the other end of C₂.

Refer to the schematic below for a visual representation:

Sallen-Key High-Pass Filter Schematic


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Verifying the Filter Design

Simulation and Testing

To ensure the filter meets the desired specifications, simulate the circuit using software tools like SPICE or Multisim. Verification steps include:

Frequency Response: Confirm that the -3 dB cut-off occurs precisely at 5 kHz.
Phase Shift: Ensure the phase shift meets the theoretical expectations at various frequencies.
Gain Verification: Check that the passband gain remains stable and within expected limits.

Adjust component values if necessary to fine-tune the filter performance based on simulation results.

Final Component Values

Component	Value	Standard Part
R₁	3.2 kΩ	3.2 kΩ Resistor
R₂	3.2 kΩ	3.2 kΩ Resistor
C₁	10 nF	10 nF Capacitor
C₂	10 nF	10 nF Capacitor
Op-Amp	TL072	TLC072CP

Circuit Implementation and Layout

Practical Considerations

Power Supply Decoupling: Place decoupling capacitors (e.g., 0.1 μF) close to the op-amp power pins to reduce noise.
Component Placement: Arrange components to minimize signal path lengths and avoid interference.
Grounding: Ensure a solid ground plane to prevent ground loops and reduce noise.

Proper layout enhances the filter’s performance by minimizing parasitic inductances and capacitances that could adversely affect the frequency response.

Advanced Considerations

Enhancing Filter Performance

Higher-Order Filters: For steeper roll-off rates beyond the cut-off frequency, consider cascading multiple second-order filters.
Component Quality: Use precision resistors and low-tolerance capacitors to maintain filter accuracy over temperature and time.
Temperature Stability: Implement temperature compensation techniques if operating in environments with significant temperature variations.

Conclusion

Designing a second-order Butterworth active high-pass filter with a 5 kHz cut-off frequency involves selecting the appropriate topology, accurately calculating component values, and meticulously assembling the circuit. The Sallen-Key configuration offers a balanced combination of simplicity and performance, making it an excellent choice for such applications. By adhering to the outlined design principles and verifying the filter through simulation and testing, one can achieve a robust and reliable high-pass filter that meets the desired specifications.

References

Design a 2nd Order High Pass Butterworth Filter with a Cutoff Frequency of 5 kHz
Active High Pass Filter - Electronics Tutorials
Chegg: Design an Op Amp Based Second Order High Pass Butterworth Filter
EEE Guide: Second Order High Pass Butterworth Filter Derivation
Texas Instruments: Single-supply, 2nd-order, Multiple Feedback High-Pass Filter Circuit
Electrical Technology: Types of Active High Pass Filter
Electronics Tutorials: Second Order Filters
Analog Devices: Basic Linear Design - Chapter 8