# Design Equations: Isolated Dielectric Resonators

## Literature survey and assumptions

Equations below have been extracted from reference: [K.1] It is aimed to summarise design equations, which give directly the resonant frequency of isolated cylindrical DRAs. There are several numerical methods which help to define the geometry of such radiators, but in practice one is interested in using equations, at least valid within a reasonable approximation, to speed up such calculation.

Below, are quoted the resonant frequencies of the most commonly used radiating modes in dielectric resonators. An alternative source for such design equations is also provided by Kishk in [Kishk, Glisson 'Bandwidth Enhancement for Split CDRAs’ Progress in Electromagnetic Research, PIER 33, 2001, pp. 97-118] based on a numerical method as shown in [F.2].

There is an article for Mode moneclature, which can inform you about indexing modes.

## Cylindrical Dielectric Resonator: Design Equations

The equations assume a cylindrical DRA of radius ‘a’, thickness ‘2H’ and dielectric constant 'εr' suspended in free space. Such an arrangement is equivalent with a cylindrical DR of radius 'a' and thickness 'H' placed above a ground plane, i.e. a Cylindrical DRA.

Free space wavenumber:

(c: speed of light, fo: resonant frequency)

Re-arranging for the resonant frequency:

__Resonant Frequencies__

Mode HEM11 (valid for 0.4 ≤ a/H ≤ 6 )

Wavenumber:

Or equivalently:

Mode TE01δ (valid for 0.33 ≤ a/H ≤ 5 )

Wavenumber:

Or equivalently:

Mode TM01δ (valid for 0.33 ≤ a/H ≤ 5 )

Wavenumber:

Or equivalently:

** **

Mode TE011+δ (valid for 0.33 ≤ a/H ≤ 5 )

Wavenumber:

Or equivalently:

## Excel file design calculator of fundamental modes

### CORRECTION: Please note, that I accidentally labelled mode TE011+δ as TM011+δ. This has now been corrected. However, the equation that calculates frequency of mode TM011+δ has been correct and it was not affected by the erroneous mode labelling. Thank you!

I have incorporated the above design equations into an excel file (correct and validated!) to quickly calculate the resonant frequencies of the modes.

I have taken every care to make sure that there are no errors in the calculations. I have cross checked the results with those found on the resonant modes of an isolated cylindrical dielectric resonator. However, please e-mail me if you notice that this is not the case! Thanks.

See below a snapshot of my excel calculator based on the above equations. It also includes a graphical tool for locating the resonant frequencies.