Sallen-Key LPF frequency scaling factor

[gehan_s]

Hi all,

I am trying to obtain Butterworth, Chebyshev and Bessal LPF responses with the Sallen-Key topology. I found this app note http://www.ti.com/lit/an/sloa049b/sloa049b.pdf and it is very helpful. According to the document if I want a 2nd order Bessal LPF, I just use the equations on top of page 9 with table 2 (in the same page). It is quite straight forward.

My questions are am I in the correct path and what is the effect of the frequency scaling factor (FSF)?
BTW the FSF is 1 for Butterworth but for Bessel and Chebyshev it has an effect. Why is that?

Regards   

[amspire]

The FSF is the relationship of the resonant frequency (for 2nd order filter) or the pole (for a first order filter) compared to the designed corner frequency of the filter.

In a Buttworth filter, and the stages are tuned for the same frequency, and only the Q changes.

In other filter types, the stages are all tuned for different frequencies and you can see this from the passband waveform. The Butterworth filter has no ripples in the passband. The Chebyshev has a flatter passband, but the passband has ripples caused by each of the different resonant frequencies.

The Bessel filter will have some ripple but it is not as obvious as most of the stages have a low Q and the resonant frequency is outside the passband. The only stages with a moderate Q are a long way outside the passband.

When you are building filters with an order greater then 2, the fact that you have stages with different Q is important when you come to select the order to put the second order Sallen-Key stages in the circuit. The can be put in any order you choose.

If you are filtering a signal with a high amplitude, you often want to put the lower Q stages first and the high Q stages last to minimize the peak signal amplitude to reduce the chance of a high Q stage clipping. If the signal level is small enough so there is no chance of clipping, you will probably order the stages to maximize the signal level in the filter.

[gehan_s]

Thank you very much for the reply amspire !!!!!!!!!!!!!

I have a small question. When you say

the stages are all tuned for different frequencies

does this mean that if i'm designing say a 5th order Bessel filter with 1kHz cutoff, I should not set the cutoff of all 5 stages as 1kHz ?

Another thing, can't I assume all resistors and capacitors (apart from the gain resistors) are equal as the document suggests?

]Thank you

[amspire]

I have a small question. When you say

the stages are all tuned for different frequencies

does this mean that if i'm designing say a 5th order Bessel filter with 1kHz cutoff, I should not set the cutoff of all 5 stages as 1kHz ?

You cannot. If all the stages have the same frequency, it is not a Bessel filter. You use the frequencies defined by the formula for the Bessel filter - or Bessel filter tables or filter design programs.

Another thing, can't I assume all resistors and capacitors (apart from the gain resistors) are equal as the document suggests?

No, you cannot.

In fact the biggest job in designing a filter is to keep tweaking the filter parameters, the cuttoff frequency and the component values for the best fit to commercially available component values.

You usually optimize for convenient capacitor values first, and if you can get away with all capacitors being the same value, then you have done brilliantly.

Once you have solved for the R and C values for a filter, you can easily shift them a little. If you increase the Cs by 10%, you have to reduce the Rs by 10%. You do not have to touch the gain resistors. For this operation, you can fiddle with the values in each stage separately.

You can also shift the  corner frequency a little without having to resolve. Assuming you have adjusted the capacitors so as many as possible are a convenient value like 1nF, you can then shift the resistors up and down by a fixed percentage until the resistor values have a best fit with a commercial value. You want to do this operation to the whole filter - so if you increase all the resistors by 1% in one stage, do the same 1% increase the the resistors in all filter stages. You will want the best matches in the high Q stages and you can tolerate the biggest errors in the low Q stages.

Try and avoid any really high Q stages. Common capacitors are only about 5% tolerance and you want to be able to use them and not expensive high tolerance capacitors. Better then 1% is hard to get. The higher the Q, the higher the tolerance of capacitors that you need.

[gehan_s]

Thank you verymych amspire for giving such an informative answer. It was very helpful !!!!!!!!!!