Electronics > Projects, Designs, and Technical Stuff

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I unfortunately found a couple of errors in the first draft, so it is withdrawn.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

**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.

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