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How to Calculate Component Values for a 3rd-Order Butterworth Low-Pass Filter

The 3rd-order Butterworth filter is a favorite among engineers because it strikes a perfect balance: it offers a steeper roll-off (18 dB per octave) than simpler filters while maintaining a “maximally flat” response in the passband. This means no annoying ripples in your signal—just a clean, smooth transition.

Here is how to calculate the component values for an active Sallen-Key realization. 1. Define Your Specifications Before touching a calculator, you need two key values: Cutoff Frequency ( ): The point where the signal power drops by half (-3dB). Capacitance (

): It’s easiest to pick a standard capacitor value first (like 10nF or 100nF) and calculate the resistors to match. 2. Understand the Filter Constants

A 3rd-order Butterworth filter is typically built by cascading a 1st-order stage and a 2nd-order Sallen-Key stage. To get the “Butterworth” response, we use specific coefficients derived from complex polynomials: (Stage 1): 1.000 (Stage 2): 1.000 3. Calculate the Resistors For a 3rd-order filter where all capacitors (

) are equal, the resistor values are determined by the following formulas: Stage 1 (The Passive RC Section) This stage uses a single resistor ( R1cap R sub 1

R1=12πfcCcap R sub 1 equals the fraction with numerator 1 and denominator 2 pi f sub c cap C end-fraction Stage 2 (The Active Sallen-Key Section) This stage uses two resistors ( R2cap R sub 2 R3cap R sub 3 ). For a standard unity-gain Butterworth configuration:

R2=12πfcC×1.392cap R sub 2 equals the fraction with numerator 1 and denominator 2 pi f sub c cap C cross 1.392 end-fraction

R3=12πfcC×0.718cap R sub 3 equals the fraction with numerator 1 and denominator 2 pi f sub c cap C cross 0.718 end-fraction

Note: The coefficients 1.392 and 0.718 ensure the proper “Q” factor for the 3rd-order response. 4. Practical Example Let’s say you want a cutoff frequency ( ) of 1 kHz using 10nF capacitors. Stage 1 Resistor ( R1cap R sub 1 ):

R1=12π×1000×10-8≈15.9kΩcap R sub 1 equals the fraction with numerator 1 and denominator 2 pi cross 1000 cross 10 to the negative 8 power end-fraction is approximately equal to 15.9 space k cap omega Stage 2 Resistor ( R2cap R sub 2 ):

R2=15.9kΩ1.392≈11.4kΩcap R sub 2 equals the fraction with numerator 15.9 space k cap omega and denominator 1.392 end-fraction is approximately equal to 11.4 space k cap omega Stage 2 Resistor ( R3cap R sub 3 ):

R3=15.9kΩ0.718≈22.1kΩcap R sub 3 equals the fraction with numerator 15.9 space k cap omega and denominator 0.718 end-fraction is approximately equal to 22.1 space k cap omega 5. Pro Tips for Success

Tolerance Matters: Use 1% metal film resistors and 5% (or better) C0G/NP0 capacitors. Small deviations can shift your cutoff frequency or introduce ripples.

Op-Amp Choice: Ensure your Op-Amp has a Gain Bandwidth Product (GBP) at least 100 times higher than your to avoid distortion.

Standard Values: You likely won’t find an 11.4 kΩ resistor. Use the closest standard E24 or E96 value, or use a potentiometer if the precision is critical.