# Reflectionless Filters for Generalized Transmission Functions

## TL;DR

This paper introduces a new theorem for designing reflectionless filters with zero reflection at all frequencies, demonstrating a practical sixth-order low-pass filter with excellent return loss and theoretical advantages over classical methods.

## Contribution

It presents a novel theorem for single-ended, reflectionless filter topologies, extending the coupled-ladder approach to pseudo-elliptic responses and comparing it favorably to traditional methods.

## Key findings

- Constructed a sixth-order reflectionless filter with pseudo-elliptic response.
- Achieved return loss better than 20 dB across pass-band, transition, and stop-band.
- Theoretical predictions closely matched measured results.

## Abstract

Single-ended circuit topologies, and a theorem for the development thereof, are presented with which one may realize constant-resistance (or reflectionless) filters, having ideally zero reflection coefficient at all frequencies and from all ports, suitable for elliptic and pseudo-elliptic filter responses. The proposed theorem produces topologies of a type known as the coupled-ladder, which has been previously studied for only polynomial responses (e.g. Butterworth, Chebyshev, etc.). A comparison between these topologies and another classical approach known as the economy bridge reveals that those proposed here have a number of theoretical and practical advantages. The theory is tested by the construction of a sixth-order, low-pass reflectionless filter exhibiting a pseudo-elliptic frequency response. Measured results are in excellent agreement with theory, and show return loss better than 20 dB throughout the pass-band, the transition-band, and up to two octaves into the stop-band.