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

Cuthbert Nyack

Here Newton's method is used to find component values for passive low pass Elliptic filters. The method works less satisfactory than it does for all pole filters. For equal terminations the same problem occurs, the derivatives become very small and the algorithm tends to jump around the solution rather than converge onto it.
In addition the method has difficulty converging as the modular angle q increases and the slope of the transition band increases.
Another feature is that the number of arithmetic operations for each iteration and the number of iterations required for convergence increases rapidly with the order of the filter.
A 5th order low pass filter is shown below. The components of this filter would be described as RS, C1, L2, C2, C3, L4, C4, C5, RL. The parallel combination L2-C2 and L4-C4 are for realizing the zeros in the stopband. Even order elliptic filters cannot be realized by RLC circuits without a transformation to move one of the zeros to infinity.
Because of this only odd order filters are examined here.

Fn = 1 to 2 shows values for 3rd order.

1, RS = 1.0, 1.1 « RL « 10.0, 3.0 « r « 25.0. 1.0 « q « 30.0.
2, RL = 1.0, 0.1 « RS « 0.95, 3.0 « r « 25.0. 1.0 « q « 30.0.

Fn = 3 to 4 shows values for 5th order.

3, RS = 1.0, 1.2 « RL « 10.0, 5.0 « r « 25.0. 4.0 « q « 45.0.
4, RL = 1.0, 0.1 « RS « 0.95, 5.0 « r « 25.0. 4.0 « q « 45.0.

Fn = 5 to 6 shows values for 7th order.

5, RS = 1.0, 1.1 « RL « 10.0, 5.0 « r « 25.0. 10.0 « q « 63.0.
6, RL = 1.0, 0.1 « RS « 0.95, 5.0 « r « 25.0. 10.0 « q « 66.0.

Fn = 7 to 8 shows values for 9th order.

7, RS = 1.0, 1.1 « RL « 10.0, 5.0 « r « 25.0. 15.0 « q « 56.0.
8, RL = 1.0, 0.25 « RS « 0.95, 5.0 « r « 25.0. 15.0 « q « 55.0.

Fn = 9 to 12 can be used to search for other solutions.

Fn = 13 to 16 can be used to examine the sensitivity of the transfer function to changes in the component values.

Image below shows a ninth order Elliptic filter. Reflection coefficient r = 20.0, Pass band ripple = 0.177dB, Modular Angle q = 43.0, Normalized transition BW = 0.466rad/s, Stop Band attenuation > 100dB, Normalized RS = 1.0, RL = 2.0, Normalized L C components shown in yellow.
Denormalized components for 3dB freq = 10kHz, Impedance scaling factor of 100.0 are shown in green.
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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