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.
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
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\$.
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.
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.
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
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.
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,
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.
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.
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).
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.
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
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.
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
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
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 actually GBW = G* BW * Q²
The 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.
