M-derived Filter - Derivation

Derivation

The building block of m-derived filters, as with all image impedance filters, is the "L" network, called a half-section and composed of a series impedance Z, and a shunt admittance Y. The m-derived filter is a derivative of the constant k filter. The starting point of the design is the values of Z and Y derived from the constant k prototype and are given by

where k is the nominal impedance of the filter, or R₀. The designer now multiplies Z and Y by an arbitrary constant m (0 < m < 1). There are two different kinds of m-derived section; series and shunt. To obtain the m-derived series half section, the designer determines the impedance that must be added to 1/mY to make the image impedance Z_iT the same as the image impedance of the original constant k section. From the general formula for image impedance, the additional impedance required can be shown to be

To obtain the m-derived shunt half section, an admittance is added to 1/mZ to make the image impedance Z_iΠ the same as the image impedance of the original half section. The additional admittance required can be shown to be

The general arrangements of these circuits are shown in the diagrams to the right along with a specific example of a low pass section.

A consequence of this design is that the m-derived half section will match a k-type section on one side only. Also, an m-type section of one value of m will not match another m-type section of another value of m except on the sides which offer the Z_i of the k-type.