Constant K Filter - Derivation

Derivation

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

Thus, k will have units of impedance, that is, ohms. It is readily apparent that in order for k to be constant, Y must be the dual impedance of Z. A physical interpretation of k can be given by observing that k is the limiting value of Z_i as the size of the section (in terms of values of its components, such as inductances, capacitances, etc.) approaches zero, while keeping k at its initial value. Thus, k is the characteristic impedance, Z₀, of the transmission line that would be formed by these infinitesimally small sections. It is also the image impedance of the section at resonance, in the case of band-pass filters, or at ω = 0 in the case of low-pass filters. 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,

.