Constant k filter

They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components.

However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

[4] The filters were designed by Campbell for the purpose of separating multiplexed telephone channels on transmission lines, but their subsequent use has been much more widespread than that.

The factor of two is introduced for mathematical convenience, since it is usual to work in terms of half-sections where it disappears.

Likewise, the image impedance of a mid-shunt section is designated ZiΠ due to the "Π" topology.

The image impedance of the half-section is dissimilar on the input and output ports: on the side presenting the series element it is equal to the mid-series ZiT, but on the side presenting the shunt element it is equal to the mid-shunt ZiΠ .

The building block of constant k filters is the half-section "L" network, composed of a series impedance Z, and a shunt admittance Y.

[10] For example, the pictured low-pass half-section has Elements L and C can be made arbitrarily small while retaining the same value of k. Z and Y however, are both approaching zero, and from the formulae (below) for image impedances, The image impedances of the section are given by[11] and Given that the filter does not contain any resistive elements, the image impedance in the pass band of the filter is purely real and in the stop band it is purely imaginary.

Above the cut-off frequency, the transmission parameters are:[11] The presented plots of image impedance, attenuation and phase change correspond to a low-pass prototype filter section.

Further additions of half-sections to either of these section forms a ladder network which may start and end with series or shunt elements.

Constant k low-pass filter half section. Here inductance L is equal Ck 2
Constant k band-pass filter half section.
L 1 = C 2 k 2 and L 2 = C 1 k 2
Image impedance Z iT of a constant k prototype low-pass filter is plotted vs. frequency . The impedance is purely resistive (real) below , and purely reactive (imaginary) above .
The transfer function of a constant k prototype low-pass filter for a single half-section showing attenuation in nepers and phase change in radians .
Gain response, H( ω ) for a chain of n low-pass constant-k filter half-sections.