# On methods to determine bounds on the Q-factor for a given directivity

## TL;DR

This paper develops convex optimization methods to determine lower bounds on the Q-factor for antennas with specified directivity, enhancing understanding of bandwidth limitations for small, arbitrarily shaped antennas.

## Contribution

It introduces semi-definite relaxation techniques to find the minimum Q-factor for total directivity, applicable to arbitrary antenna shapes and directions, improving previous bounds.

## Key findings

- SDR provides tight lower bounds on Q-factor for given directivity.
- The method applies to arbitrary shaped antennas and other non-convex constraints.
- Results are validated with full EM-simulations of high-directivity antennas.

## Abstract

This paper revisit and extend the interesting case of bounds on the Q-factor for a given directivity for a small antenna of arbitrary shape. A higher directivity in a small antenna is closely connected with a narrow impedance bandwidth. The relation between bandwidth and a desired directivity is still not fully understood, not even for small antennas. Initial investigations in this direction has related the radius of a circumscribing sphere to the directivity, and bounds on the Q-factor has also been derived for a partial directivity in a given direction. In this paper we derive lower bounds on the Q-factor for a total desired directivity for an arbitrarily shaped antenna in a given direction as a convex problem using semi-definite relaxation techniques (SDR). We also show that the relaxed solution is also a solution of the original problem of determining the lower Q-factor bound for a total desired directivity. SDR can also be used to relax a class of other interesting non-convex constraints in antenna optimization such as tuning, losses, front-to-back ratio. We compare two different new methods to determine the lowest Q-factor for arbitrary shaped antennas for a given total directivity. We also compare our results with full EM-simulations of a parasitic element antenna with high directivity.