Low Pass Butterworth Filter with Passive Components and Unequal Terminations Applet.

Cuthbert Nyack

Here Newton's method is used to find component values for passive low pass Butterworth filters. The method works for unequal terminations. For equal terminations, the derivatives become very small and the algorithm tends to jump around the solution rather than converge onto it. Fortunately analytic expressions are available for odd and even Butterworth equiterminated filters so this is not a problem.
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 12 shows values for 3rd order.

Fn = 1,2 shows the values found in tables.
3 shows the equiterminated case.
4, RL = 1.0, RS = 1.001 to 1.006.
5, RL = 1.0, RS = 1.001 to 4.2.
6, RL = 1.0, RS = 4.0 to 11.0.
7, RL = 1.0, RS = 1.001 to 4.2.
8, RL = 1.0, RS = 4.0 to 11.0.
9 to 12 shows that the algorithm also converges on the values found by impedance scaling.
9, RL = 1.0, RS = 1.2 to 10.0.
10, RL = 1.0, RS = 1.001 to 1.5.
11, RL = 1.0, RS = 0.05 to 0.95.
12, RL = 1.0, RS = 0.94 to 0.999.

Fn = 13 to 22 shows values for 4th order.

13 to 15 shows values found in tables.
13, RL = 1.0, RS = 1.02 to 3.5.
14, RL = 1.0, RS = 2.4 to 12.0.
15, RL = 1.0, RS = 1.005 to 1.5.
16 shows the equiterminated case.
17, RL = 1.0, RS = 1.0005 to 4.2.
18, RL = 1.0, RS = 4.0 to 11.0.
19, RL = 1.0, RS = 2.6 to 12.0.
20, RL = 1.0, RS = 1.001 to 2.5.
21, RL = 1.0, RS = 1.02 to 12.0.

Fn = 23 to 33 shows values for 5th order.

23 and 24 shows values found in tables.
23, RL = 1.0, RS = 0.07 to 0.99.
24, RL = 1.0, RS = 0.94 to 0.9999.
25 shows the equiterminated case.
26, RL = 1.0, RS = 1.02 to 4.2.
27, RL = 1.0, RS = 4.2 to 12.0.
28 to 30 shows values from impedance scaling.
28, RL = 1.0, RS = 0.07 to 0.99.
29, RL = 1.0, RS = 1.03 to 4.2.
30, RL = 1.0, RS = 4.0 to 12.0.
31, RS = 1.0, RL = 0.07 to 0.99.
32, RS = 1.0, RL = 1.01 to 4.2.
33, RS = 1.0, RL = 4.2 to 10.0.

Fn = 34 to 38 shows values for 6th order.

34 and 35 shows values found in tables.
34, RL = 1.0, RS = 1.01 to 4.2.
35, RL = 1.0, RS = 4.0 to 11.0.
36 to 38 shows other values.
36, RL = 1.0, RS = 1.01 to 4.2.
37, RL = 1.0, RS = 4.0 to 12.0.
38, RS = 1.0, RL = 0.07 to 0.98.

Fn = 39 to 51 shows values for 7th order.

39 shows values found in tables.
39, RL = 1.0, RS = 1.07 to 0.99.
40 shows the equiterminated case.
41, RL = 1.0, RS = 1.01 to 4.2.
42, RL = 1.0, RS = 4.0 to 10.0.
43, RS = 1.0, RL = 0.07 to 0.99.
44, RS = 1.0, RL = 1.01 to 4.2.
45, RS = 1.0, RL = 4.0 to 10.0.
46 to 50 show other solutions.
46, RL = 1.0, RS = 0.07 to 0.99.
47, RL = 1.0, RS = 0.1 to 0.99.
48, RL = 1.0, RS = 0.1 to 0.99.
49, RL = 1.0, RS = 0.1 to 0.99.
50, RL = 1.0, RS = 0.1 to 0.99.
51 shows values from impedance scaling.
51, RL = 1.0, RS = 0.1 to 0.99.

Fn = 52 to 56 shows values for 8th order.

52 shows the equiterminated case.
53 and 54 shows values found in tables.
53, RL = 1.0, RS = 1.01 to 4.2.
54, RL = 1.0, RS = 4.0 to 10.0.
55 and 56 shows other solutions.
55, RL = 1.0, RS = 1.01 to 4.2.
56, RL = 1.0, RS = 4.0 to 10.0.

Fn = 57 to 64 shows values for 9th order.

57 and 58 shows values found in tables.
57, RL = 1.0, RS = 0.07 to 0.98.
58, RL = 1.0, RS = 0.97 to 0.999.
59 shows the equiterminated case.
60 and 64 shows other solutions.
60, RL = 1.0, RS = 1.01 to 4.2.
61, RL = 1.0, RS = 4.0 to 10.0.
62, RL = 1.0, RS = 0.07 to 0.99.
63, RS = 1.0, RL = 1.01 to 4.2.
64, RS = 1.0, RL = 4.0 to 10.0.
Fn = 65 to 71 can be used to search for other solutions.
Fn = 72 to 78 can be used to examine the sensitivity of the transfer function to changes in the component values.

Image below shows a ninth order Butterworth filter. Normalized RS = 0.6667, RL = 1.0, Normalized L C components shown in yellow.
Low pass denormalized components for 3dB freq = 7kHz, 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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