Electronic – Transfer Function of Bessel Filter

filtertransfer function

the Bessel filter transfer function is defined via bessel polynomials. If we consider for example a 2nd order filter, the transfer function is:
$$ H(s) = \frac{3}{s^2+3*s+3} $$
I wanted to build a simulation for such a filter with a Sallen-Key-Architecture. Therefore I consulted this design guide by TI.
They define the transfer function of a 2nd order low pass as:

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Ao is 1 since I want the gain to be unity. I looked at the table below in order to correctly calculate the C- and R-values.

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Hence the transfer function becomes:
$$ H(s) = \frac{1}{0.618*s^2+1.3617*s+1} $$

I ran the simulation and looked at the bode plot. It showed the desired result (the -3db cutoff frequency was as calculated).

However I do not understand why the transfer function looks so differently. Its definetely not a Bessel polynomial. I checked the step response and observed an overshoot of 0.4% as one would expect for a Bessel filter. Therefor I have 3 Questions:

  1. How come that the transfer function in the ti design guide is not a bessel polynomial.
  2. Should the pole location of a 2nd order Bessel filter be the same for any filter with a certain cutoff frequency?
  3. Can a second order bessel low pass have a different Q factor than 0.5773?

Thanks!

Best Answer

How come that the transfer function in the ti design guide is not a bessel polynomial.

Let's look at the transfer function you have written: -

\$H(s) = \dfrac{1}{0.618s^2+1.3617s+ 1}\$

Rearranging: -

\$H(s) = \dfrac{1.6181}{s^2+2.2034s+ 1.6181}\$

The equation is now in standard form : \$H(s) = \dfrac{\omega_n^2}{s^2+2\zeta\omega_ns+ \omega_n^2}\$

And clearly \$\omega_n\$ = \$\sqrt{1.6181}\$ hence 2.2034/\$\sqrt{1.6181}\$ = 1.732. This bit is important because it is \$\sqrt3\$.

For a Bessel 2nd order low pass filter 2\$\zeta\$ = \$\sqrt3\$ hence zeta is 0.866.

enter image description here

Picture source

In the picture I've manipulated R to give me a damping ratio (zeta) of precisely 1.732 - look at the peak in the step response - 1.00433 volts - exactly right for Bessel. Look at the phase delay plotted on the upper graph - maximally flat and gradually becoming 90 degrees at the natural resonant frequency. Fd (the damped frequency) is precisely 0.5 - also indicative of Bessel.

Can a second order bessel low pass have a different Q factor than 0.5773?

0.5773 is the reciprocal of \$\sqrt3\$ and no it has tto be that Q for a Bessel LPF.

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