# Are distributed element filters really distributed?

When looking at distributed filters etched into PCB, I have a hard time understanding why they called distributed. Most of them can be broken into sections and have basically a 1 to 1 correspondence with lumped element filters.

How are distributed filters really distributed?

• Lumped element filters can be viewed as a single object for analysis. If the time between the filters (based on rise / fall times or phase in a microwave world) then they are independent and cannot be analysed as a single object. That means they are distributed. Feb 2, 2021 at 9:37

In distributed element filters the circuit isn't put together of discrete capacitors nor inductors. The visible elements are pieces of transmission lines which are coupled together by joints and couplings where the wave jumps over a gap form one line to another.

They are said to be distributed because transmission lines (at least those which work carrying TEM waveform) can be seen as having capacitance, inductance and loss resistance uniformly distributed along the line.

You seemingly have an idea. "It's not really distributed because it contains discrete parts, no matter those parts may be modeled with distributed quantities!" The idea is valid, but it's not in accordance with our established conventions.

One way to design distributed element filters is to use Richard's Transformation. It's a math trick (see NOTE1) which helps to convert LC filters to transmission line circuit filters. LC ladder circuits will be transformed to transmission line ladders. Read this for a start: https://en.wikipedia.org/wiki/Commensurate_line_circuit

But there are also other design methods which are based initially on wave carrying circuits, not on lumped element LC filters.

NOTE1: R's transformation belongs to frequency variable transformations. The idea is to change the frequency variable in the transfer function to get a filter which can be built from different kind of parts or to convert the transfer function to different filtering type; for ex. from lowpass to bandpass. The simplest widely used frequency variable transformation is to change f to 1/f. It changes a lowpass filter to a highpass filter. In a LC filter circuit that means changing inductors to capacitors and vice versa.

Distributed element filters are called that to distinguish them from lumped element filters.

In a lumped element filter, assuming it was practically possible to have zero-size components, then in theory you could build the filter in zero size, electrical length is not required for it to function.

In a distributed element filter, some, or often all, of the reactive components use their electrical length as an important parameter.

Often, series elements of a low pass filter will be 'thin' lines, having higher impedance than the system impedance, so looking inductive. In a hand-waving way you can understand how they might work as inductors. However, to understand accurately the effect they have on the response of the filter, you need to factor in their electrical length. A special electrical length is λ/4, this transforms a short at one end to an open at the other.

A common low pass filter construction is a through transmission line with a number of 'capacitive' shunt elements to ground at λ/4 spacings. For accuracy, these are often built as 'bow-ties', so their connection point to the line can be accurately defined. The length of these bow-ties is more of an important parameter than their total (capacitive) area. Although this filter looks like a cascade of inductor/capacitor sections to the casual eye, it still works with a system-impedance through line, and relies on the impedance transformation of each quarter wavelength section.