SLOA049 Application note

SLOA049D July 2000 – February 2023

B.2 Sixth-Order Low-Pass Bessel Filter

Referring to Table 9-2, for a sixth-order Bessel filter, we can write the required circuit transfer function as:

Equation 29.

H_{LP} (f) = \frac{K}{(- {(\frac{f}{1 .6060 f_{c}})}^{2} + 1 .2202 \times \frac{j f}{f_{c}} + 1) (- {(\frac{f}{1 .6913 f_{c}})}^{2} + 0 .9674 \times \frac{j f}{f_{c}} + 1) (- {(\frac{f}{1 .9071 f_{c}})}^{2} + 0 .5124 \times \frac{j f}{f_{c}} + 1)}

Figure 15-2 shows an MFB circuit implementation and the required component values. $f_{c}$ is the –3-dB point. The overall gain of the circuit in the pass band is $K = K_{a} \times K_{b} \times K_{c}$ .

Figure 15-2 Sixth-Order Low-Pass Filter Topology Cascading Three MFB Stages

Stage	$f_{c}$	$Q$	$K$
1	$\frac{1}{1.6060 \times 2 π \sqrt{R_{2 a} R_{3 a} C_{1 a} C_{2 a}}}$	$\frac{\sqrt{R_{2 a} R_{3 a} C_{1 a} C_{2 a}}}{R_{3 a} C_{1 a} + R_{2 a} C_{1 a} + R_{3 a} C_{1 a} (- K_{a})} = 0.5103$	$K_{a} = \frac{- R_{2 a}}{R_{1 a}}$
2	$\frac{1}{1.6913 \times 2 π \sqrt{R_{2 b} R_{3 b} C_{1 b} C_{2 b}}}$	$\frac{\sqrt{R_{2 b} R_{3 b} C_{1 b} C_{2 b}}}{R_{3 b} C_{1 b} + R_{2 b} C_{1 b} + R_{3 b} C_{1 b} (- K_{b})} = 0.6112$	$K_{b} = \frac{- R_{2 b}}{R_{1 b}}$
3	$\frac{1}{1.9071 \times 2 π \sqrt{R_{2 c} R_{3 c} C_{1 c} C_{2 c}}}$	$\frac{\sqrt{R_{2 c} R_{3 c} C_{1 c} C_{2 c}}}{R_{3 c} C_{1 c} + R_{2 c} C_{1 c} + R_{3 c} C_{1 c} (- K_{c})} = 1.0234$	$K_{c} = \frac{- R_{2 c}}{R_{1 c}}$