# How to design a 4th/6th/8th passive low pass filter?

I am trying to design a 4th/6th/8th passive low pass filter. The filter is used after a DAC to generate a fixed frequency sine wave. The DAC will export a square wave at 1kHz. So the pass band of the LP filter will be 1.5kHz with stopband at 3kHz.

There are various information on the internet to implement an active lowpass filter. The reason for us to use passive filter is to avoid noises brought by OpAmps and reduce the number of OpAmps at the same time to reduce cost. Of course a voltage follower will be applied at the output of the passive filter.

So here comes my question.. How to design a 4th/6th/8th passive low pass filter?

• If your primary signal is being generated by a DAC, why not output a 1kHz sine wave from the DAC in the first place? It would be a stepped sine of course, but the the noise components would be lower amplitude and higher frequency than the square wave so your passive filter would be more effective (and/or could be lower order). – markt May 22 '14 at 11:18
• Thanks for your reply. We could only afford 8points/cycle. Our goal is to 1) generate 1kHz signal 2)avoid 2kHz signal 3) others do not matter much. When we use stepped sine wave, the output produces 2kHz component as well. It may be caused by glitches of DAC. And we suspect that it is caused by stepped sine as well.. Is our suspect correct? – richieqianle May 23 '14 at 4:28

Your reasoning not to use an op-amp is basically flawed. here I quote you: -

Of course a voltage follower will be applied at the output of the passive filter.

This voltage follower can in fact be made into a 2nd order low pass filter using the sallen key topology: -

As you can see the op-amp is configured as a voltage follower (unity gain) but overall, due to the feedback of C1, the circuit behaves as a 2nd order low pass filter.

Another point is also largely flawed: -

The reason for us to use passive filter is to avoid noises brought by OpAmps

Using passive components to achieve a ~1.5kHz high order low pass filter is really problematic for two reasons. Firstly, the Q of the circuit - if you want good performance from the filter then RC filters are OUT - they will not achieve a decent enough Q to get the stop-band performance likely to be needed so you have to use LCR filters BUT given the operating frequency, the size of the inductor is going to be problematic and it will be far more costlier than a basic half-decent op-amp. I'm talking dollars versus cents here.

And secondly, if you could get away with RC filters, in order to get anywhere near a decent performance, the 1st stage values of R and X$_C$ will need to be significantly smaller than the 2nd stage values in order not to excessively load the 1st stage with the input impedance of the 2nd stage. This means high values of resistor and high noise.

Two RC stages get you a 2nd order. In order to achieve 8th order you'll need 8 stages. Let's say the first stage used 100 ohms and each successive stage used a resistor that was only 3 times bigger (5x would be better). 300 ohms for the 2nd stage, 900 ohms for the 3rd stage, 2k7 for the 4th, 8k1 for the 5th, 24k3 for the 6th, 72k9 for the 7th and 218k7 for the 8th - that final resistor will produce 2.3 uV RMS of noise across a 1.5kHz bandwidth. A typical op-amp having 10nV/sqrt(Hz) noise will produce 0.39 uV RMS across the same bandwidth.

My extremely confident advice is forget about passive filters and use op-amps. Use several "voltage followers" configured as sallen key filters. Link to great calculator.

If you want to prove to yourself that a 2nd order low pass filter made from R and C cannot achieve a Q greater than 0.5 here is a calculator: -

• Yes - the link as given by Andy aka leads us to a good filter design program. However, one should know that the unity gain S&K filter is only one option of various S&K alternatives. It has - for example - the disadvantage that both capacitors must NOT be equally chosen. Other filter design programs allow to select a gain of two for the fixed gain amplifier. This can be realized very easily with two equal resistors in the negative feedback path - and has the advantage that now both capacitors are allowed to be equal. – LvW May 22 '14 at 8:14
• @LvW In the type of work that I do I see what you suggest as a disadvantage. I use a lot of cascaded SK filters on jobs and I made the decision ages ago to stick with unity gain filters and if I needed a gain stage this would be separate (mainly because tweaking the gain to suit a particular job didn't mean swapping out capacitors to re-adjust the filter characteristics prior to the gain change). Horses for courses. – Andy aka May 22 '14 at 8:29
• and 218k7 for the 8th - that final resistor will produce 2.3 uV RMS of noise across a 1.5kHz bandwidth How did you calculated the noise? – m.Alin May 22 '14 at 9:44
• @m.Alin I used this: daycounter.com/Calculators/Thermal-Noise-Calculator.phtml – Andy aka May 22 '14 at 9:47
• @Andy aka: It was my only intention to point to the fact that unity gain S&K is not the only one option which is available. As you know, it is always good knowing some options - and, in particular, for designing filters we often have to face some conflicting requirements. Nevertheless, what do you think is the main "disadvantage" of a gain-of-two topology? – LvW May 22 '14 at 10:20

A passive filter design of higher order should be made with the help of filter tables (to be found in specialized filter handbooks) or preferably with the help of suitable filter design programs. Such programs are available for free - for example: AADE filer design (www.aade.com). This program offers several toplogy alternatives up to a filter order of n=16.

• thanks for your reply! I was looking for a software.. However the one you provided seem to generate RLC LP filters only. Any choices with only RC filters? – richieqianle May 22 '14 at 6:59
• No - as mentioned by Andy aka, passive RC filters have very poor selectivity. It is rather uncommon to use passive RC combinations for orders above n=4. – LvW May 22 '14 at 8:04
• What about for a 4th order passive lp filter? – richieqianle May 23 '14 at 4:33
• @richieqianle: Yes - it is possible, of course. However (as mentioned already by Andyaka) due to loading problems the resistor values do increase from stage to stage which could rise problems. Thus, it is wise to use (at least) one decoupling buffer in the middle (or even between all stages). – LvW May 23 '14 at 10:18
• To use two voltage followers in a 4th order passive lpf design, then there would be no point of using a passive one Im afraid.. Generally how to design a 4th order passive LPF? – richieqianle May 23 '14 at 12:47