# Active low-pass filter design

I have read TI's application note Active Low-Pass Filter Design a couple of times, particularly page 19.

Section A.1.1 shows some simplified steps for designing a filter by choosing m and n as a function of gain K and Q.

1. Although it is probably explained in the paper, I still don't understand the meaning of m and n and what they represent.

2. I don't understand how the tables on page 12 were derived and, especially, what they are used for.

I think these two questions are critical during the design of a filter.

• See page 13 of the document you linked to. m & n are used to describe the type of filter (Butterworth, Bessel, Chbyshev). Also see section 9 where the coefficients are listed.
– qrk
Commented Dec 29, 2023 at 20:50
• It's there on page 19, the page you refer to: Letting R1 = mR , R2 = R , C1 = C , and C2 = nC Commented Dec 29, 2023 at 20:50

As said by others in the comments, numbers m and n are YOUR choices to attempt to get at least some of the resistors and capacitors available as standard off the shelf components such as resistors 680 Ohm, 1kOhm, 2.2 kOhm, etc... The formulas do not give absolute values for the resistor and capacitor values, they only define their ratios, so you can also make some component value selections without changing the filtering properties. Numbers n and m are those that you can play with in given limits to get the needed components more easily with less parallel or serial combinations.

The tables:

Filtering transfer functions are approximations of ideal brickwall filtering transfer functions, which pass certain frequency range as is and stop all other frequencies completely.The approximations are developed by mathematicians who developed the general approximation theory. Some of them, for ex. guys like Chebyshev or Bessel surely never thought electronic filters, because such things didn't exist. Some good approximations are originally developed just for filtering - like Cauer's elliptical approximations.

The approximation theory gives the optimal s-plane rational functions when the order (the highest power of s) of the rational function is given at first. In what sense a function is optimal? It varies. Butterworth, Chebyshev, Bessel and Cauer approximations optimize different things and sacrifice some others.

Filter tables in filter handbooks give the filters calculated for limit frequency fc=1Hz. The component values must be scaled for the actual fc.

Your attached application note constructs higher order lowpass filters by cascading 2nd order filter stages. It's mathematically proven to be a good way to limit the needed component value accuracy. By cascading two 2nd order filters instead of making directly a single opamp 4th order filter allows much more inaccurate resistors and capacitors for the same performance accuracy.

The tables are presented so that you select the filter order and pick Q and FSF for each filter stage from the table and calculate the needed component values with the formulas. You input the actual fc, one component value for each stage and the selectable factors n and m.

Deriving the tables is possible only for those who understand the approximation theory math and know there the unique fundamental optimization ideas of Butterworth, Chebyshev and Bessel approximations. Prepare to study a couple of years engineering math to be up to the task. The theory is presented in academic level network synthesis theory books. Engineering handbooks have only ready to use tables with no proofs.

ADDED after the questioner wondered in a comment when should he select Sallen-Key filter or what are the reasons to select multi-feedback filter. And which facts tell the right selection between Butterworth, Chebyshev and Bessel?

Sallen and Key developed inductor-free filters in the era when modern high-gain opamps were not available. Sallen-Key circuit topology works acceptably even with single transistor or single tube voltage follower amps, no high gain opamp is a must. The drawback is the limitation of the max.available passband gain. Multi-feedback filter topology has not that limitation. Other topologies are known with their own pros and cons. One general factor to select a topology is the performance when the frequency is so high that the opamp non-ideality starts to be notable. Another is the sensitivity to component value inaccuracies.

Butterworth, Tshebyshev, Bessel and Elliptic (by Cauer) are not about the circuit schematic at all, but what's the rational s-plane function used to approximate the ideal brickwall filter. The approximations are totally different in the next areas:

• how steep is the transition from the passband to the stopband with given filter order
• how large are the gain variations vs. frequency in the passband and in the stopband
• how much differently different frequencies get delayed in the filter

The 3rd property is extremely essential in some applications. For ex. data communication gets trashed if signal pulses are distorted to a ringing mess by the frequency dependent delay variations. It's easy to show that the frequency dependent delay causes problems even easier than frequency dependent passband gain variations. Bessel's approximation is good when the delay for passband signals must be quite constant, but other approximations have steeper transition from passband to stopband. The common term for non-frequency dependent delay is "phase linearity". The price of getting phase linear filtering by selecting Bessel's approximation is a must to use higher order filter for certain frequency selectivity.

All of the mentioned things are told in filter theory textbooks. Filter engineering handbooks can have poor theory content, because they assume the designer knows what he wants.

Circuit analyzer programs make possible to test easily how well filters separate the passband and the stopband and how badly they distort the pulses. Math programs make it possible with no schematic by evaluating what the transfer function causes. Such programs did not exist when complex communication and radar equipment became vital (WW2). The only way to get something complex to work was to make all on theory-based calculations with extremely care.

• @KaleM maybe try using this calculator: sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm if you want to design a 2nd order low-pass filter. Scroll down the page to find this section: Calculate the R and C values for the Sallen-Key filter at a given frequency and Q factor. For Butterworth, choose a q-factor of 0.7071 and enter your cut-off frequency. Then press the calculate button. Commented Dec 30, 2023 at 11:43
• @KaleM I inserted a little more of potentially interesting things. Commented Dec 30, 2023 at 17:33
1. Although it is probably explained in the paper the meaning of m and n ... I still don't understand what they are and what they represent :/

The value for $$\n\$$ is selected:

1. Specifically so that the capacitor ratio plays nice with capacitor values as they are only found in a few standard values; and,
2. So that it is on the steep part of the $$\m\$$ vs $$\n\$$ curve such that the usual capacitor variations (which will impact the actual ratio you get when parts are stuffed) do not materially impact the $$\m\$$ used for the resistor ratio.

So, the values of $$\m\$$ and $$\n\$$ aren't uniquely determined by the filter type. They are, instead, bounded by the filter type.

The author of the TI literature (or the original author used as a source to develop it) made choices. And these choices had implications. But be certain that the specific pairings weren't a necessary consequence of the mathematics itself. Instead, rather by a human taking into consideration practical facts about electronic parts and variations and selecting a choice for one and accepting the necessary consequence to the other. They are good choices, but not the only.

For example, a Butterworth could also use $$\n=4.7\$$ and $$\m=0.1377\$$ (the exact value I got was $$\0.137697373888625\$$) and it would still be a Butterworth. There is freedom in choice here. But the TI paper doesn't cover that. Instead, it makes a choice for you.

If I understand your question, then I touch on the topic here, using Butterworth and Bessel to explain. There's a desire to scale the frequency so that the various polynomial types have their $$\-3.0103\:\text{dB}\$$ point where a designer would expect them rather than where they naturally fall using the canonical form. If you follow the argument I make there then you can replicate their table values. (Or should be able to. I've not gone and done it for you. But I think I know the process to follow in order to do so.)
That said, they are used in scaling the frequency so that the cross-over is at $$\-3.0103\:\text{dB}\$$ (as probably expected by readers) and setting the shape of each stage (where a $$\Q\$$ is given for 2nd order cases.) The linked answer here includes enough of a discussion that you should see how higher order filters are broken down into these kinds of tables specifying only 1st order and 2nd order stages.