# Slopes of band-pass op-amp

I have the following double band-pass op-amp. The cut-off frequencies are 34 and 50kHz.

How can I calculate the slope of the gains around the cut-off frequencies? I want to know at which frequency I have -3dB.

• That is an absolutely horrible way to draw a schematic. Take a look at electronics.stackexchange.com/questions/28251/…. Oct 18, 2014 at 10:19
• Why have you used two single order BPFs cascaded? For the same number of components, you could have cascaded a 2nd order LP and a 2nd order HP filter. This makes the results much more easily found on the internet and is a more standard solution and gives better performance without crappy interactions between the LP and HP parts of the filters you have. Oct 18, 2014 at 10:32
• The -3 dB points of each of your stages are set at 33.9 kHz and 49.8 kHz. This means that when you cascade two stages, the gain at these frequencies will be -6 dB, and your -3 dB bandwidth is going to be much narrower. Oct 18, 2014 at 11:04

It looks like the upper (low pass) 3dB point for 1 of the cascaded filters is 50kHz based on the values R = 47k and C = 68p. In fact it works out at 49.798 kHz. For the same reasoning, the lower (high pass) 3dB point will be 33.863kHz. However, there is a problem. Take a look at this frequency response for a normalized LPF: -

Clearly, the 3dB point is at F = 1 (or 50kHz in your example) but at the lower cut-off frequency (shown as a grey vertical line at about F = 0.6 (or 30kHz in your example) there is still some attenuation of about 1.5dB and at 34kHz the attenuation will be about 2dB.

This means that what you thought were your 3dB points for just one of your cascaded circuits are in fact the 5dB points. Both parts of one filter interact with each other and muddy the waters so you have to go back to basics and re-evaluate what you want.

I would suggest going for two 2nd order filters; one a low pass and one a high pass. Take a look at the roll-off for a normalized 2nd order LPF: -

If you pick the value of Q correctly you can ensure that the lower (HP) 3dB point is largely unaffected by the upper 3dB point. Now you can design a two stage cascaded filter that comprises a 2nd order LPF and an independant 2nd order HPF. The formulas for each independant stage are "carved in stone" and ditto the 3dB points. There will be hardly any interaction of upper 3dB point and lower 3dB BY DESIGN!

Here is a link to a 2nd order (sallen key) low pass filter design tool.

Also check the same site out for a high pass sallen key filter too.