Elliptic filter

General information

The elliptic filters , also called filters of Cauer , in homage to the theorist who in exhiba the first the interest, are filters whose answer is characterized by an undulation as well in Band-width as in stop-band. Cauer showed that they are optimal in the sense that no filter, to order given, presents a stiffer cut only the elliptic filters. Mathematically, these filters call upon the formalism of the transformations in conformity , they are thus based on the theory of the elliptic functions of Jacobi, from where them name.

Contrary to the other filters, which present to the maximum two degrees of freedom (order and possibly undulation), the elliptic filters have three degrees of freedom: their order, the undulation in band-width and stiffness of the cut, which also determines the minimal attenuation in stop-band. In the tables, they thus appear in the form DC N ρ θ , where N is the order, ρ is the undulation and θ the angle of cut (stiffness): θ = 90° corresponds to a band of null transition, and with θ = 0° one finds a filter of Tchebychev of the type 1. To ρ = 0, one returns to a filter of Tchebychev of the type 2.

The figure opposite assistance to better including/understanding how the parameters play. All the filters are of order 5. The curves in red, blue and green correspond all three to filters having 1 dB of undulation in band-width, but to angles of cut respectively of 30,45 and 60°. One sees that the more important the angle is, the more the filter is selective: the attenuation increases more quickly, but it also reaches a maximum with lower values. The curve mauve corresponds to a filter of undulation 0,1 dB and angle 45°. The poles (frequencies where the attenuation becomes infinite) are the same ones as the blue curve (of the same filter angle but of undulation 1 dB), but the final attenuation obtained is less important, which is the price to pay to have a weaker undulation, with constant order.

The elliptic filters of an odd nature have impedances of identical entry and exit. The filters of an even nature are classified in two categories: the sub-type B has impedances of entry and of exit different, the sub-type C has equal impedances.

These filters, just like the filters of Tchebychev of order 2, present a topology which alternates the circuits LLC and the simple components (L or C). They are not ladder-type filters.

The use of the elliptic filters remains rather confidential, because of the difficulty inherent in their calculation. This disadvantage from now on is raised by the use of programmes of synthesis per computer.

Optimization

It is possible to carry out at least two simple but useful types of optimization on the elliptic filters:

If one wishes to eliminate a particular frequency or a narrow band around a given frequency, one can place the first pole of attenuation precisely at this frequency. This thus guarantees an almost infinite attenuation an elimination of the frequency, and fixes the angle θ. It thus remains to exploit the order and the undulation to determine the other characteristics of the filter.
the antiresonant circuits series which carry out the poles of attenuation can be placed in any order within the filter. This introduced a light freedom on the realization which can make it possible to find combinations maximizing or minimizing the value of critical components.

See too

Internal bonds

linear Filtre
Filtre of Bessel
Filtre of Butterworth
Filtre combs from there
Filtre of Legendre
Filtre of elliptic Tchebychev
Filtre

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