Analog Filters: a Compilation of Standard Transfer Functions (UPDATED 2) - Page 1
Forum section: electronics / projects

author: Benta
author_rank: Super Contributor
timestamp: January 22, 2021, 08:15:36 pm
content: Analog filters, whether active or passive, always follow a standard transfer function, normally in the complex s-domain alias LaPlace notation.I was not able to find a comprehensive list of all the general transfer functions, so I decided to create one myself.I offer it here for you to use as you wish in your personal work and designs, or to pin it on the wall, or...If you want to use the equations in your own document, each one is available in .odf format (open document formula). PM me.I appreciate feedback, I've carefully checked the equations for errors, but many eyes see better than just two.UPDATE 2:The PDF has been updated and now includes band-pass and band-stop functions as well (peak/notch only).A new spreadsheet is included (.ZIP) if you want to play with second-order gain, cutoff frequency, damping ratio etc.Cheers.

author: jonpaul
author_rank: Super Contributor
timestamp: February 02, 2021, 11:18:07 am
content: The theory was done at Bell Labs, 1920s.Many fine books on filter design and theory are available, suggest to search them.Kind Regards,Jon

author: mawyatt
author_rank: Super Contributor
timestamp: February 02, 2021, 06:39:39 pm
content: Nice work.For specific filter types, like Butterworth, Bessel, Chebyshev, Cauer and so on, Zverev's Handbook of Filter Synthesis is a great resource.Best,Edit: If you like to play around with active filters, here's a simple equal value R and C, third order Butterworth 1/[(S+1)(S^2 + S +1)].https://www.eevblog.com/forum/beginners/filter-topology-sallen-key-and-mfb-resistor-order-matter-based-on-their-values/msg3403234/#msg3403234

author: Cerebus
author_rank: Super Contributor
timestamp: February 02, 2021, 07:18:56 pm
content: And for those who want the same in MathJax/\$\LaTeX\$ format (which, as should be evident, can be directly rendered on the forum):Low Pass: \$F(s) = K \cdot \frac{1}{s+\omega_0}\$High Pass: \$F(s) = K \cdot \frac{s}{s+\omega_0}\$All Pass: \$F(s) = K \cdot \frac{s-\omega_0}{s+\omega_0}\$Low pass even order: \$F(s)=K \cdot \frac{1}{s^2+2\zeta_a\omega_{0_a}s+\omega_{0_a}^2} \cdot ... \cdot \frac{1}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$Low pass odd order: \$F(s)=K \cdot \frac{1}{s+\omega_{0_a}} \cdot \frac{1}{s^2+2\zeta_b\omega_{0_b}s+\omega_{0_b}^2} \cdot ... \cdot \frac{1}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$High pass even order: \$F(s)=K \cdot \frac{s^2}{s^2+2\zeta_a\omega_{0_a}s+\omega_{0_a}^2} \cdot ... \cdot \frac{s^2}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$High pass odd order: \$F(s)=K \cdot \frac{s}{s+\omega_{0_a}} \cdot \frac{s^2}{s^2+2\zeta_b\omega_{0_b}s+\omega_{0_b}^2} \cdot ... \cdot \frac{s^2}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$All pass even order: \$F(s)=K \cdot \frac{s^2-2\zeta_a\omega_{0_a}s+\omega_{0_a}^2}{s^2+2\zeta_a\omega_{0_a}s+\omega_{0_a}^2} \cdot ... \cdot \frac{s^2-2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$All pass odd order: \$F(s)=K \cdot \frac{s-\omega_{0_a}}{s+\omega_{0_a}} \cdot \frac{s^2-2\zeta_b\omega_{0_b}s+\omega_{0_b}^2}{s^2+2\zeta_b\omega_{0_b}s+\omega_{0_b}^2} \cdot ... \cdot \frac{s^2-2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$Band pass: \$F(s)=K \cdot \frac{s}{s^2+2\zeta_a\omega_{0_a}s+\omega_{0_a}^2} \cdot ... \cdot \frac{s}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$Band stop: \$F(s)=K \cdot \frac{s^2+\omega_{0_a}^2}{s^2+2\zeta_a\omega_{0_a}s+\omega_{0_a}^2} \cdot ... \cdot \frac{s^2+\omega_{0_n}^2}{s^2+2\zeta_n\omega_{0_n}s+\omega_{0_n}^2}\$If you want to use any of these on the forum just 'quote' this post and cut and paste the relevant bit.I used Benta's pdf as a crib. I was more engaged in getting the rendering right, so if anyone spots any maths mistakes shout out and I'll correct them - you tend to go a bit cross-eyed after more than a couple of equations in \$\LaTeX\$.

author: Benta
author_rank: Super Contributor
timestamp: February 02, 2021, 08:14:00 pm
content: Thank You for your feedback, I feel encouragedThere's nothing revolutionary about this, it's simply about creating a one-stop place for this stuff.Attached is a new document on multiple-feedback (MFB) and finite-gain ("Sallen-Key") filters.It lists the generic transfer functions for the two topologies in admittance and impedance forms. "Generic" in this case means, that no low/high/all/band pass functions are involved, but only the basic transfer functions of the circuits themselves.Plugging the normal LaPlace component equivalents into the equations give the classic transfer functions, but the fun part is the option to use more complex admittances or impedances, thus either modelling non-ideal parts or creating special filter transfer functions.Cheers.

author: Nominal Animal
author_rank: Super Contributor
timestamp: February 06, 2021, 12:49:43 am
content: Nice! I do the same for oddball math (and molecular potential models) I use. For whatever odd reason, I (also) like to use LibreOffice and its Math module, even though I can do LaTeX just as well. I guess my preference is because in that "mode", I want to keep focus on the math and how it looks, as opposed to on how to best express it; and I'm much better when I focus on one thing at a time. (Just like I like to doodle on pen and nice thick paper, when thinking out ideas. Now, writing an article, that's a different kettle of fish altogether.)It's Laplace, by the way, not LaPlace, although thedudewas French: Pierre-Simon, marquis de Laplace.

author: mawyatt
author_rank: Super Contributor
timestamp: February 07, 2021, 05:19:06 pm
content: I've always wondered why some of Heaviside's work gets lumped into the Laplace domain and he gets little recognition? His algebraic solutions of differential equations related to early transmission lines preceded the use of Laplace Functions I believe, he also formulated Maxwell's original equations into the 4 we now refer as "Maxwell's Equations". I have always felt he got shortchanged by history, as so few know about his pioneering work.Best,

author: RFdesigner
author_rank: Newbie
timestamp: March 07, 2021, 11:18:13 pm
content: Zverev's book sits behind me on the bookshelf. Not so easy to find unless they reissued it. My copy was given to me by my boss when he retired many years ago. Cheers!

author: mawyatt
author_rank: Super Contributor
timestamp: March 08, 2021, 04:49:04 pm
content: There's the original Zverev Hardcopy and later a Softcopy was produced, did have both but lent the Hardcopy that wasn't returned!!Best,

author: Dunckx
author_rank: Contributor
timestamp: March 28, 2021, 10:28:14 pm
content: I have just discovered that Zverev's "Filter Synthesis" book is out of copyright and may be downloaded from the Internet Archive for free!https://ia803101.us.archive.org/20/items/HandbookOfFilterSynthesis/Handbook%20of%20Filter%20Synthesis.pdfHTH

author: RoGeorge
author_rank: Super Contributor
timestamp: April 19, 2021, 10:14:49 am
content: Quote from: Dunckx on March 28, 2021, 10:28:14 pmGreat handbook I didn't know before, thank you!It even shows the "N-Path mixer"/"PolyPhase Mixer" at page 35/586, "Fig.1.33 The Digital Filter", recently discussed inhttps://www.eevblog.com/forum/rf-microwave/polyphase-or-n-path-mixer/Quote from: Zverev's "Filter Synthesis"

author: mawyatt
author_rank: Super Contributor
timestamp: April 26, 2021, 03:08:48 pm
content: Yes, even Zverev referred to this commutating filter, which even predates Zverev's classic reference book by a few years!! However, these Discrete Time Continuous Amplitude (DTCA) filter concepts for other uses evaded general discovery until the Tayloe's Detector in the late 90s and then the PolyPhase, or N-Path Mixers ~2008. We did employ the DTCA commutating filter (as well as DTCA Chirp-Z techniques for Real Time SA) techniques in the early 80s for a narrow band tunable RF filter, but it took another 20~30 years for "other" uses of this very powerful technique to emerge!Best,

author: teletypeguy
author_rank: Contributor
timestamp: July 29, 2021, 06:36:22 pm
content: For just getting to work on a filter without all the synthesis, Analog Devices has an awesome tool:https://tools.analog.com/en/filterwizard/Define low/high/bandpass and passband/stopband needs, and boom you get a schematic. Not only can you view mag/phase... you can specify cap and resistor tolerances and see the sensitivity envelope. The recommended opamps are ADI, of course, but you can look up the gain-bandwidth and such to determine substitutions for each stage. Plug it all into ltspice and fine tune. ADI make great components, though I am more partial to ti opamps as they are usually cheaper than adi for the same performance, and cmos opamps (eg: opa2376) are finally getting into the realm of really low noise for low-power/low-volt applications.

author: KRISTOFFER
author_rank: Contributor
timestamp: September 26, 2022, 08:16:52 pm
content: I had a play with this a year back for use in my guitar amplifier. Without getting into the maths, if you want an active steep slopped high pass, low pass or bandpass filter you will get to a point where there is an unwanted peak just before the slope. In audio, the peak looks bad on paper but to the human ear it is pretty much undetected. In my case it was to attenuate everything below 120 hz before any distortion. But, a bit like any classic guitar amplifier with trebble middle and bass controls, you will be on for ever tweaking it then listening to it and chasing your tail.

author: precaud
author_rank: Frequent Contributor
timestamp: October 13, 2022, 12:06:00 pm
content: Quote from: mawyatt on February 02, 2021, 06:39:39 pmConfession: That is the only book I have ever stolen from a library and never returned (40 years ago).

author: srb1954
author_rank: Super Contributor
timestamp: February 10, 2023, 08:54:46 pm
content: Quote from: precaud on October 13, 2022, 12:06:00 pmZverev's book can be downloaded now so you can return the hard copy version to the library.

author: seamusdemora
author_rank: Regular Contributor
timestamp: February 16, 2023, 04:32:00 am
content: Quote from: precaud on October 13, 2022, 12:06:00 pmThe bad news for you is that your former librarian is now a moderator here. lol

author: tggzzz
author_rank: Super Contributor
timestamp: February 16, 2023, 09:57:04 am
content: Quote from: mawyatt on April 26, 2021, 03:08:48 pmAs I've notedelsewhere, I somehow disinterred a1960BSTJ paper and in 1979-81used itto make a narrowband filter to remove noise.The concepts and references you raised in your otherthreadaremostinteresting. Thanks.

author: seamusdemora
author_rank: Regular Contributor
timestamp: February 17, 2023, 04:13:13 am
content: Quote from: tggzzz on February 16, 2023, 09:57:04 amWould you care to make a few remarks regarding the potential utility of these filters toward a superior front end for aWWVBreceiver?WWVB is a narrow-band, LF (60 kHz), AM signal broadcast from Ft. Collins, CO. It is similar to theMSFservice in the UK). One of the issues with WWVB is the amount of man-made noise in this LF band. I know that a filter and preamp can improve the S/N ratio, but I've no experience in filter designs (since university). The filtersynthesizer I triedcame up with what appears to me to be a complex design (10th order!?). I wondered if there wasn't a "better way", and when I read this post I thought, "Maybe this is it?".Best Rgds,~S

author: tggzzz
author_rank: Super Contributor
timestamp: February 17, 2023, 10:07:38 am
content: Quote from: seamusdemora on February 17, 2023, 04:13:13 amI suggest you read the references in mawyatt's other thread, since they will be much more considered than my response.https://www.eevblog.com/forum/rf-microwave/polyphase-or-n-path-mixer/My application required the removal of wideband white noise, not typical out-of-band RF signals. Having said that, yes I think they would be worth investigating, particularly in conjunction with other standard techniques.Overall it is an interesting topic, one that deserves its own thread, and one that shouldn't divert this "references" thread

author: mick_lee
author_rank: Newbie
timestamp: March 27, 2023, 02:49:43 pm
content: Why would you do this? Anatoly Zerev did this in 1967!https://www.amazon.co.uk/Handbook-Filter-Synthesis-Anatol-Zverev/dp/0471749427

author: TonyStewart
author_rank: Contributor
timestamp: November 14, 2023, 03:45:46 pm
content: I recall having used variable Q with a constant peak gain and tuneable f BPFs in my grad thesis in the mid-70s and discovered a relationship not taught in school about GBW is not G * BW in active filters.I discovered that the required GBW in active filters is actuallyGBW = G* BW * Q²The Q² significantly challenges Op Amps, especially with Q's = 100. Fortunately, GBQ = 100 MHz Op Amps do exist and a couple more than this. I have validated this using the TI Analog filter application, which used to be downloadable but shows the ideal and actual response with the required GBW and Q of each stage.

author: hneve
author_rank: Regular Contributor
timestamp: June 07, 2024, 12:07:16 pm
content: That sounds like a really useful resource you've created! It's great to see you sharing it with the community. I'm sure many people will find it valuable for their projects. If I may offer a suggestion, maybe consider sharing it in a more accessible format like a Google Doc or a GitHub repository to make it easier for others to access and contribute.

author: Jesse_G_EE
author_rank: Newbie
timestamp: June 14, 2024, 04:30:55 pm
content: Here's a good resource I've used in the passed for an ultrasonic radar project:http://sim.okawa-denshi.jp/en/Fkeisan.htm

author: joseangel
author_rank: Contributor
timestamp: June 05, 2025, 05:36:33 pm
content: Quote from: mick_lee on March 27, 2023, 02:49:43 pmTHE book. It has saved me from losing hair a few times, recently for designing some very high bandwidth analog delay linesHere you can find a more permanent link & content,https://archive.org/details/HandbookOfFilterSynthesis

author: joseangel
author_rank: Contributor
timestamp: June 05, 2025, 05:39:14 pm
content: Another very good reference, free, from Analog Devices.This one is much more basic than Zverev, however most common topics are covered or at least mentioned, good starting point for learning,https://www.analog.com/media/en/training-seminars/design-handbooks/basic-linear-design/chapter8.pdf

author: Terry Bites
author_rank: Super Contributor
timestamp: July 04, 2025, 07:48:02 am
content: I snapped up a copy of the Zverev bible of filters on eBay. I paid about €50, a bargain.They often change hands for serious cash:www.ebay.co.uk/sch/i.html?_nkw=zverev+filters&_sacat=0&_from=R40&_trksid=m570.l1313&_odkw=zverev&_osacat=0Mine used to belong to a company called CTI, developers of early video cassettes.Have I bought stolen goods?Put it on your ebay watch list.I got a lot of mileage from a compact package called NuHertz.Coilcraft offer a neutered version for free.www.coilcraft.com/en-us/other/coilcraft-lc-filter-designer-software/?srsltid=AfmBOoomVBWJYkQsiDeLAap-9x3D-TpijoFWUu5nwMpAGywv5GqgrllE

author: ballsystemlord
author_rank: Frequent Contributor
timestamp: July 04, 2025, 06:17:29 pm
content: Quote from: Terry Bites on July 04, 2025, 07:48:02 amAre you talking about "Handbook of Filter Synthesis, Paperback by Zverev, Anatol I." or "Filtering in the Time and Frequency..., Zverev, Anatol "?Thanks!

author: mawyatt
author_rank: Super Contributor
timestamp: July 05, 2025, 01:59:11 pm
content: Quote from: Terry Bites on July 04, 2025, 07:48:02 amMaybe!! Don't lend out your version, they have a habit of evaporatingWe had our hardback "borrowed" by an employee long ago, then left the company and didn't return the book. This left us with only the paperback version which we still have.Wonderfull filter resource indeedBest