Author Topic: Analog Filters: a Compilation of Standard Transfer Functions (UPDATED)  (Read 9517 times)

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Offline Benta

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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I unfortunately found a couple of errors in the first draft, so it is withdrawn.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 2nd 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.

 
« Last Edit: March 20, 2021, 11:20:10 pm by Benta »
 
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Offline jonpaul

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #1 on: February 02, 2021, 11:18:07 am »
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
Jon Paul
 

Offline mawyatt

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #2 on: February 02, 2021, 06:39:39 pm »
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
« Last Edit: February 02, 2021, 07:48:10 pm by mawyatt »
Research is like a treasure hunt, you don't know where to look or what you'll find!
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Offline Cerebus

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #3 on: February 02, 2021, 07:18:56 pm »
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\$.
Anybody got a syringe I can use to squeeze the magic smoke back into this?
 
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Offline Benta

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #4 on: February 02, 2021, 08:14:00 pm »
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.

« Last Edit: February 02, 2021, 10:15:18 pm by Benta »
 

Online Nominal Animal

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #5 on: February 06, 2021, 12:49:43 am »
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.
 

Offline mawyatt

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Re: Analog Filters: a Compilation of Standard Transfer Functions
« Reply #6 on: February 07, 2021, 05:19:06 pm »
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, 
Research is like a treasure hunt, you don't know where to look or what you'll find!
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Offline RFdesigner

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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!
 
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Offline mawyatt

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

Best,
Research is like a treasure hunt, you don't know where to look or what you'll find!
~Wyatt Labs by Mike~
 

Offline Dunckx

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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
"God help us, we're in the hands of engineers." - Dr. Ian Malcolm, Jurassic Park
 
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Online RoGeorge

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Re: Analog Filters: a Compilation of Standard Transfer Functions (UPDATED)
« Reply #10 on: April 19, 2021, 10:14:49 am »
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/

Quote from: Zverev's "Filter Synthesis"
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.


 
 
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Offline mawyatt

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Re: Analog Filters: a Compilation of Standard Transfer Functions (UPDATED)
« Reply #11 on: April 26, 2021, 03:08:48 pm »
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,
« Last Edit: April 26, 2021, 03:11:19 pm by mawyatt »
Research is like a treasure hunt, you don't know where to look or what you'll find!
~Wyatt Labs by Mike~
 


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