# Parametric Amplifier Matching Using Legendre Prototypes

## TL;DR

This paper introduces Legendre polynomial-based matching networks for Josephson parametric amplifiers, offering lower ripple and smoother roll-off compared to Chebyshev designs, with practical synthesis methods and prototype examples.

## Contribution

It presents a novel approach using Legendre polynomials for JPA matching networks, including prototype coefficients and synthesis techniques.

## Key findings

- Lower ripple and gentler roll-off than Chebyshev networks.
- Prototype design achieving 20dB gain at 5GHz with 500MHz bandwidth.
- Comprehensive tabulation of prototype coefficients for various parameters.

## Abstract

In this note we describe Josephson parametric amplifier (JPA) matching networks based on Legendre polynomials. These networks typically exhibit lower ripple and gentler roll-off than Chebyshev networks with similar parameters, and can be viewed as bridging the gap between Butterworth and Chebyshev ones. We tabulate prototype coefficients for parametric amplifiers based on Legendre polynomials with a range of gain and ripple parameters, and for a range of network orders. We also use this opportunity to further illustrate the synthesis of these networks based on methods from previous work, and synthesize a prototype JPA with 20dB gain at a center frequency of 5GHz with a bandwidth of 500MHz.