EEVblog Electronics Community Forum
Electronics => Projects, Designs, and Technical Stuff => Topic started by: Benta on January 22, 2021, 08:15:36 pm

Analog filters, whether active or passive, always follow a standard transfer function, normally in the complex sdomain 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I unfortunately found a couple of errors in the first draft, so it is withdrawn.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 bandpass and bandstop 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.

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

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/filtertopologysallenkeyandmfbresistorordermatterbasedontheirvalues/msg3403234/#msg3403234 (https://www.eevblog.com/forum/beginners/filtertopologysallenkeyandmfbresistorordermatterbasedontheirvalues/msg3403234/#msg3403234)

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^22\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^22\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^22\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^22\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 crosseyed after more than a couple of equations in \$\LaTeX\$.

Thank You for your feedback, I feel encouraged :)
There's nothing revolutionary about this, it's simply about creating a onestop place for this stuff.
Attached is a new document on multiplefeedback (MFB) and finitegain ("SallenKey") 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 nonideal parts or creating special filter transfer functions.
Cheers.

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 (https://en.wikipedia.org/wiki/Laplace) was French: PierreSimon, marquis de Laplace.

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,

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!

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

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

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 (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 "NPath mixer"/"PolyPhase Mixer" at page 35/586, "Fig.1.33 The Digital Filter", recently discussed in
https://www.eevblog.com/forum/rfmicrowave/polyphaseornpathmixer/ (https://www.eevblog.com/forum/rfmicrowave/polyphaseornpathmixer/)
A method having the dimensions of a genuine breakthrough is shown in Fig. 1.33. Here the mechanical commutators, shown schematically, would be replaced in practice by transistor diode analoggates 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 previously 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.
(https://www.eevblog.com/forum/projects/analogfiltersacompilationofstandardtransferfunctions/?action=dlattach;attach=1212691;image)

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 NPath Mixers ~2008. We did employ the DTCA commutating filter (as well as DTCA ChirpZ 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,