# Cut-off frequency and -3dB frequency of Butterworth filter

I am in the process of designing a Butterworth low pass filter in Sallen-Key topology. I use the Analogue Devices Filter design tool.

After reading the TI application report I have calculated all the values base on the formulas provided. Fc Cut-off frequency (frequency at which phase shift is equal to 90°). Q quality factor by sliding between Chebyshev to Bessel. K is gain converted from dB. I have noticed that the capacitor ratio is fixed as 10. Sensible to select the ratio of caps rather than resistors. Still unsure why 10 was selected. Perhaps getting any series of caps in the one-decade ratio is easy.

https://www.ti.com/lit/an/sloa024b/sloa024b.pdf

I simulated the circuit in LTSpice.

I have noticed about a 70Hz difference between the point at which the phase shifts by 90° and the -3dB point. Therefore I am interested in how to calculate the relation between both. When Filter parameters are selected in Filter Design Tool the frequency, in this case, 2KHz refers to -3dB. However, LTSpice simulation shows that at 2KHz the phase shift is equal to 90° and the signal is attenuated by 2.7dB. I also used an online calculator to verify. (Please note that C1 and C2 notation is inversed in the TI document compared to the Okawa-denshi calculator). Sallen-Key Low-pass Filter Design Tool - Result - (okawa-denshi.jp)

Lastly in the TI application report on page 4, the formula for system gain at f=fc is simplified to: H(lp)=-jKQ in case of Bessel filter Q=0.707 and K=2. Therefore, H(lp) = 1.414 = 3dB.

I am a bit puzzled as to why Filter Design Tool asks for passband frequency and then uses it as cut-off frequency.

Additionally, the formula on page 4 refers to gain not phase.

Case with ideal op-amp and greater accuracy of the components.

Simulating with an ideal op-amp and much greater precision of the components decreased the difference between -3dB and 90° phase shift point (pole of my system). I begin to suspect that my initial mismatch might be caused by rounding the components to E series components. Before I accept this as an answer perhaps someone could confirm or deny this hypothesis.

• Your title says "Butterworth", yet you keep talking about a Bessel. Which one is it? Also, the -3 dB for Bessel is not the same as the -90 deg. Apr 6 at 12:08
• @aconcernedcitizen Thank you for pointing out it is a Butterworth filter. Apr 6 at 12:52