Analog Filters: a Compilation of Standard Transfer Functions (UPDATED)

[Benta]

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:

I unfortunately found a couple of errors in the first draft, so it is withdrawn.

I've redacted the original document extensively and have added sections on how to calculate magnitude and phase response of the 2^nd order transfer functions as well.
The band-pass and band-stop equations have been removed, as they are pretty useless in practice having only resonant responses.


A little spreadsheet is included if you want to play with gain, cutoff frequency, damping ratio etc.

Cheers.

 

[jonpaul]

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

[mawyatt]

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

[Cerebus]

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\$.

[Benta]

Thank You for your feedback, I feel encouraged
There'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.

[Nominal Animal]

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 the dude was French: Pierre-Simon, marquis de Laplace.

[mawyatt]

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, 

[RFdesigner]

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!

[mawyatt]

There's the original Zverev Hardcopy and later a Softcopy was produced, did have both but lent the Hardcopy that wasn't returned!!

Best,

[Dunckx]

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.pdf
HTH

[RoGeorge]

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.pdf
HTH

Great 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 in
https://www.eevblog.com/forum/rf-microwave/polyphase-or-n-path-mixer/

A   method   having   the   dimensions   of   a   genuine   breakthrough  is  shown  in  Fig.  1.33.   Here  the  mech-anical  commutators,  shown  schematically,  would   be   replaced  in  practice  by  transistor  diode  analog-gates  driven   by  conventional   logic  circuits  at  an   angular   velocity,  co0.  The  identical  RC  circuits  have  a  lowpass  response  which  is  so  modified  by  the  sampling  action  that    VJV2    exhibits    a   related    bandpass    response    centered    at    a>0.    This   method    shows   promise    of    yielding  figures  for   Q  and  stability  which  have  pre-viously    been   obtainable    only   with   crystal   filters.   Definite  results  on  the  performance  of  this  device  will  be    available  soon,  but  it  is too  new  to  be  evaluated  at  present.


 

[mawyatt]

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,

[teletypeguy]

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.

[KRISTOFFER]

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.

[precaud]

For specific filter types, like Butterworth, Bessel, Chebyshev, Cauer and so on, Zverev's Handbook of Filter Synthesis is a great resource.

Confession: That is the only book I have ever stolen from a library and never returned (40 years ago).

[srb1954]

For specific filter types, like Butterworth, Bessel, Chebyshev, Cauer and so on, Zverev's Handbook of Filter Synthesis is a great resource.

Confession: That is the only book I have ever stolen from a library and never returned (40 years ago).

Zverev's book can be downloaded now so you can return the hard copy version to the library.

[seamusdemora]

For specific filter types, like Butterworth, Bessel, Chebyshev, Cauer and so on, Zverev's Handbook of Filter Synthesis is a great resource.

Confession: That is the only book I have ever stolen from a library and never returned (40 years ago).

The bad news for you is that your former librarian is now a moderator here.   lol

[tggzzz]

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!

As I've noted elsewhere, I somehow disinterred a 1960 BSTJ paper and in 1979-81 used it to make a narrowband filter to remove noise.

The concepts and references you raised in your other thread are most interesting. Thanks.

[seamusdemora]

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!

As I've noted elsewhere, I somehow disinterred a 1960 BSTJ paper and in 1979-81 used it to make a narrowband filter to remove noise.

The concepts and references you raised in your other thread are most interesting. Thanks.

 

Would you care to make a few remarks regarding the potential utility of these filters toward a superior front end for a WWVB receiver?

WWVB is a narrow-band, LF (60 kHz), AM signal broadcast from Ft. Collins, CO. It is similar to the MSF service 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 filter synthesizer I tried  came 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

[tggzzz]

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!

As I've noted elsewhere, I somehow disinterred a 1960 BSTJ paper and in 1979-81 used it to make a narrowband filter to remove noise.

The concepts and references you raised in your other thread are most interesting. Thanks.

 

Would you care to make a few remarks regarding the potential utility of these filters toward a superior front end for a WWVB receiver?

WWVB is a narrow-band, LF (60 kHz), AM signal broadcast from Ft. Collins, CO. It is similar to the MSF service 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 filter synthesizer I tried  came 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

I 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

[mick_lee]

Why would you do this? Anatoly Zerev did this in 1967! 
https://www.amazon.co.uk/Handbook-Filter-Synthesis-Anatol-Zverev/dp/0471749427