# Designing 4th order Butterworth Multiple feedback bandpass filter

I'm trying to design a 4th order Butterworth bandpass filter using Multiple feedback topology. The design requirements I am trying to achieve for this filter are; Q factor = 10, Av = 11, fc = 100kHz , BW = 10kHz.

In the book Op-amps for every one book pdf, I came across two design tables:

The First of the table correspond to my desired filter requirements with Q factor of 10 and filter coefficients values for the parameters a1, b1, α. And filter calculations are given as well.

The Second table shows different 4th order coefficients values for ai,bi,Qi and uses ki.

My questions are;

1) Why the two tables for 4th order Butterworth filter have different coefficients and Q factor values?

2) The coefficient parameters a1, b1, Q, α are the same as ai, bi, Qi, ki respectively?

3) Which is the appropriate table to use? If the second table is better for designing the filter. What are the calculations steps needed in using those coefficient parameters. Since no formulas are given in the book.

• I think you see two lines for 4th order because it is implemented as two 2nd order sections. What you are seeing are the damping coefficients; one for each 2nd order section. 0.765 and 1.848 would be about the right values, too. – jonk Nov 15 '18 at 19:22
• What @jonk said, plus you should take some time to read the rest of the book, it will be worth it. Who knows? Maybe you'll find other explanations in there, too. – a concerned citizen Nov 15 '18 at 19:32
• LAD_145, may I direct your attention to the fact that table 16.5 contains LOWPASS parameters!! – LvW Nov 15 '18 at 19:58
• It may motivate you to grab up more about this if I tell you that the two 2nd order damping values in that 16-5 table come from $\sqrt{2\pm\sqrt{2}\:}$, which themselves derive from $2\operatorname{cos}\left(\frac{\pi}{2N}\left[2i-1\right]\right)$, with $i=1$ and $i=2$. – jonk Nov 15 '18 at 21:25
• LAD-145....I have a question: Are you required to use two second-order bandpass stages in series (staggerd tuning)? Are you aware that there are alternative (and better !) methods for designing a 4th-order bandpass? – LvW Nov 16 '18 at 8:35

Just above Table 16-2:

...with a1and b1 being the second-order low-pass coefficients of the desired filter type. To simplify the filter design, Table 16 – 2 lists those coefficients, and provides the α values

The fourth order band-pass filter in that example is built from two partial second order filters. This method is called staggered tuning.

This means that the coefficients a1 and b1 of Table 16-2 are second order coefficients, and not fourth order coefficients.

Table 16-5. shows low-pass coefficients. Compare with Table 16-1:

Thus, you can use the Table 16-2 and staggered tuning to design your fourth order band-pass filter.

• Rrz0....why do you recommend to use table 16-1 ? Here a LOWPASS parameters listed. How should one use these values for a bandpass?? Answer: Lowpass-bandpass transformation. However, I doubt if this procedure should be recommended!! – LvW Nov 16 '18 at 9:51
• Thanks for correcting. Edited to 16-2, as stated in the book. – rrz0 Nov 16 '18 at 10:23

This design is going to deviate from expected performance due to insufficient GBW requirements.

With a gain of 3.3 in each stage with a Q of ~14 in each stage at staggered frequencies to achieve a net BW of 10% fc= 10kHz, if one truly expects a Butterworth response 1 octave up, means the minimum GBW product is >> 50 MHz. A GBW of 10MHz reduces the output 50%.