Low Pass Bessel Filter with passive components and different Terminations Applet.

Cuthbert Nyack

On this page Newton's method is used to find component values for low pass Bessel Filters with orders 3 to 9 and with different terminations.
To find the component values, the coefficients of powers of s in the theoretical and circuit transfer functions are equated. For an nth order filter, this results in n nonlinear equations.
Newton's method is then used to try to solve the nonlinear equations. The method works fairly satisfactorily with Bessel filters but not as satisfactory with other filter approximations.
A 5th order low pass filter is shown below. The components of this filter would be described as RS, C1, L2, C3, L4, C5, RL. For low pass the capacitors are connected as shunt elements and the inductors as series elements.

Component values can be found by using the applet below.
The algorithm not only converges to the values found in tables but also to other values. Although these values have the same magnitude and phase behaviour, they have different impedance characteristics. In some cases, these other values are indicated.
The function of the applet is set by Fn which is controlled by scrollbar 0.

Fn = 1 to 6 is used for 3rd order filters.

Fn = 1 shows the values that are normally found in tables for RL = 1.0 and RS from 0.07 to 1.0, here it goes to 2.2.
Fn = 2 shows values for RS from 2.0 to 10.0.
Fn = 3,4 shows other solutions found by the algorithm. In this case, the circuit is the reverse of the first.
Fn = 5,6 shows values for RS = 1 and different RL.

Fn = 7 to 11 is used for 4th order filters.

Fn = 7,8 shows values found in tables, 7 is for for RL = 1 and RS = 0.5 to 3.2, 8 is for RS = 3.0 to 10.0.
Fn = 9,10 shows another soln for RL = 1 and RS = 0.07 to 10.0.
Fn = 11 shows values for RS = 1, RL = 0.07 to 2.0.

Fn = 12 to 15 shows values for 5th order.

12(table values) RL = 1 RS = 0.07 to 3.2.
13, RL = 1, RS = 3.0 to 10.0.
14, RS = 1, RL = 0.07 to 3.2.
15, RS = 1, RL = 3.0 to 10.0.

Fn = 16 to 21 shows values for 6th order.

16, (tables) RL = 1, RS = 0.5 to 3.2.
17, RL = 1, RS = 3.0 to 10.0.
18, RL = 1.0, RS = 0.3 to 3.2.
19, RL = 1.0, RS = 3.0 to 10.0.
20, RS = 1.0, RL = 0.07 to 4.2.
21, RS = 1.0, RL = 3.0 to 9.8.

Fn = 22 to 29 shows values for 7th order.

22, (tables) RL = 1.0, RS = 0.1 to 5.0.
23, RL = 1.0, RS = 3.0 to 10.0.
Fn = 24 to 28 shows other component values found by the algorithm.
24, RL = 1.0, RS = 0.1 to 3.5.
25, RL = 1.0, RS = 0.1 to 3.5.
26, RL = 1.0, RS = 0.1 to 3.5.
27, RL = 1.0, RS = 0.1 to 3.5.
28, RS = 1.0, RL = 0.1 to 3.5.
29, RS = 1.0, RL = 3.0 to 10.0.

Fn = 30 to 35 shows values for 8th order.

30 (tables) RL = 1.0, RS = 0.5 to 3.5.
31, RL = 1.0, RS = 3.0 to 10.0.
32, RL = 1.0, RS = 0.1 to 4.2.
33, RL = 1.0, RS = 4.0 to 12.0.
34, RS = 1.0, RL = 0.1 to 3.5.
35, RS = 1.0, RL = 3.0 to 10.0.

Fn = 36 to 40 shows values for 9th order.

36 (tables) RL = 1.0, RS = 0.1 to 2.8.
37, RL = 1.0, RS = 2.4 to 7.2.
38, RL = 1.0, RS = 6.0 to 11.0.
39, RL = 1.0, RS = 0.1 to 3.2.
40, RL = 1.0, RS = 3.0 to 10.0.

Fn = 41 to 47 can be used to search for other solutions.
Fn = 48 to 54 can be used to examine the sensitivity of the transfer function to changes in the component values.

Image below shows a ninth order Bessel filter. Normalized RS = 1.0, RL = 1.0, Normalized L C components shown in yellow.
Low pass denormalized components for 3dB freq = 20kHz, Impedance scaling factor of 150.0 are shown in green. High pass denormalized components are shown in pink.
Normalized poles are shown in blue.
Nonlinear functions which must be zeroed for the circuit transfer function to be equal to the theoretical transfer function are shown in red.
Incremental changes to the circuit components after the last iteration are shown in magenta.

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