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

Cuthbert Nyack

Here Newton's method is used to find component values for passive low pass Chebyshev 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 Chebyshev equiterminated filters. Component values are not possible for equiterminated even filters with finite ripple.
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 7 shows values for 3rd order.

1, Tables, RL = 1.0, 0.07 « RS « 0.99, 0.01 « Rip « 2.5dB.
2 shows the equiterminated case.
3,4 shows other soln.
3, RL = 1.0, 1.02 « RS « 4.0, 0.01 « Rip « 2.5dB.
4, RL = 1.0, 4.0 « RS « 10.0, 0.01 « Rip « 2.5dB.
5,6,7 shows impedance scaling soln.
5, RL = 1.0, 0.07 « RS « 0.99, 0.01 « Rip « 2.5dB.
6, RL = 1.0, 1.02 « RS « 4.0, 0.01 « Rip « 2.5dB.
7, RL = 1.0, 4.0 « RS « 10.0, 0.01 « Rip « 2.5dB.

Fn = 8 to 11 shows values for 4th order.

For even order Chebyshev filters, the value of RS must be greater than 1(more accurately it must be greater than RL) for convergence. The minimum value of RS for which convergence occurs depends on the ripple.
For Rip = 0.5dB min RS ~ 2.0ohms. For Rip = 1dB min RS ~ 2.7ohms. For Rip = 1.5dB min RS ~ 3.4ohms. For Rip = 2dB min RS ~ 4.1ohms.

8, tables, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
9, 10, 11 shows other solns.
9, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
10, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
11, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.

Fn = 12 to 18 shows values for 5th order.

12, Tables, RL = 1.0, 0.1 « RS « 0.97, 0.01 « Rip « 2.5dB.
13, shows the equiterminated case.
14 to 18 shows other solns.
14, RL = 1.0, 1.2 « RS « 4.0, 0.01 « Rip « 2.5dB.
15, RL = 1.0, 4.0 « RS « 10.0, 0.01 « Rip « 2.5dB.
16, impedance scaling, RL = 1.0, 0.1 « RS « 0.97, 0.01 « Rip « 2.0dB.
17, impedance scaling, RL = 1.0, 1.1 « RS « 4.0, 0.01 « Rip « 2.0dB.
18, impedance scaling, RL = 1.0, 4.0 « RS « 10.0, 0.01 « Rip « 2.5dB.

Fn = 19 to 23 shows values for 6th order.

19, tables, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
20 to 23 shows other values found by the algorithm.
20, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
21, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
22, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.
23, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.

Fn = 24 to 31 shows values for 7th order.

24, Tables, RL = 1.0, 0.1 « RS « 0.97, 0.01 « Rip « 2.5dB.
25, shows the equiterminated case.
26 to 31 shows other values found by the algorithm.
26, RL = 1.0, 0.15 « RS « 0.97, 0.01 « Rip « 2.5dB.
27, RL = 1.0, 0.15 « RS « 0.95, 0.01 « Rip « 2.5dB.
28, RL = 1.0, 0.19 « RS « 0.95, 0.01 « Rip « 2.5dB.
29, RL = 1.0, 0.15 « RS « 0.95, 0.01 « Rip « 2.5dB.
30, RL = 1.0, 0.15 « RS « 0.92, 0.01 « Rip « 2.5dB.
31, impedance scaling, RL = 1.0, 0.16 « RS « 0.92, 0.01 « Rip « 2.5dB.

Fn = 32 shows values for 8th order.

32, tables, RL = 1.0, 1.2 « RS « 10.0, 0.01 « Rip « 2.5dB.

Fn = 33 to 37 shows values for 9th order.

33, shows the equiterminated case.
34, Tables, RL = 1.0, 0.1 « RS « 0.7, 0.01 « Rip « 2.5dB.
35, Tables, RL = 1.0, 0.7 « RS « 0.83, 0.01 « Rip « 2.5dB.
36, Tables, RL = 1.0, 0.8 « RS « 0.91, 0.02 « Rip « 2.5dB.
37, Tables, RL = 1.0, 0.8 « RS « 0.95, 0.01 « Rip « 0.01dB.

Fn = 38 to 44 can be used to search for other solutions.

Fn = 45 to 51 can be used to examine the sensitivity of the transfer function to changes in the component values.

Image below shows a ninth order Chebyshev filter with pass band ripple = 0.5dB. Normalized RS = 0.9, RL = 1.0, Normalized L C components are shown in yellow.
Low pass denormalized components for 3dB freq = 5kHz, Impedance scaling factor of 100.0 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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