# Dielectric Modelling

The dielectric properties of the dielectric is defined.

## Frequency Independent

The media is defined in terms of the relative permittivity ( ${\epsilon }_{\text{r}}$ ), relative permeability ( ${\mu }_{r}$ ), magnetic loss tangent ( $\mathrm{tan}{\delta }_{u}$ ), and the dielectric loss tangent ( $\mathrm{tan}\delta$ ) or conductivity ( $\sigma$ ).

For example, low loss dielectric substrates are typically specified in terms of the loss tangent, while human tissue (used in specific absorption rate studies) are specified in terms of conductivity.

The effective permittivity of the dielectric is given by:

(1) ${\epsilon }_{eff}={\epsilon }_{0}{\epsilon }_{r}\left(1-j\mathrm{tan}\delta \right)$
or
(2) ${\epsilon }_{eff}={\epsilon }_{0}{\epsilon }_{r}-j\frac{\sigma }{\omega }$

## Debye Relaxation

The Debye relaxation1 describes the relaxation characteristics of gasses and fluids at microwave frequencies. It has been derived for freely rotating spherical polar molecules in a predominantly non-polar background. The method is defined in terms of the relative static permittivity ( ${\epsilon }_{s}$ ), relative high frequency permittivity ( ${\epsilon }_{\infty }$ ) and the relaxation frequency ( ${f}_{r}$ ).

(3) ${\epsilon }^{*}={\epsilon }_{\infty }+\frac{{\epsilon }_{s}-{\epsilon }_{\infty }}{1+j\frac{f}{{f}_{r}}}$

## Cole-Cole

The Cole-Cole2 model is similar to the Debye model, but uses one additional parameter to describe the material. The model is defined in terms of the relative static permittivity ( ${\epsilon }_{s}$ ), relative high frequency permittivity ( ${\epsilon }_{\infty }$ ), relaxation frequency ( ${f}_{r}$ ) and the attenuation factor ( $\alpha$ ).

(4) ${\epsilon }^{*}={\epsilon }_{\infty }+\frac{{\epsilon }_{s}-{\epsilon }_{\infty }}{1+\left( j \frac{f}{{f}_{r}}{\right)}^{1-\alpha }}$

## Havriliak-Negami

The Havriliak-Negami3 is a more general model and should be able to successfully model liquids, solids and semi-solids. It is defined in terms of the relative static permittivity ( ${\epsilon }_{s}$ ), relative high frequency permittivity ( ${\epsilon }_{\infty }$ ), relaxation frequency ( ${f}_{r}$ ), attenuation factor ( $\alpha$ ) and the phase factor ( $\beta$ ).

(5) ${\epsilon }^{*}={\epsilon }_{\infty }+\frac{{\epsilon }_{s}-{\epsilon }_{\infty }}{{\left[1+\left( j \frac{f}{{f}_{r}}{\right)}^{1-\alpha }\right]}^{ \beta }}$

## Djordjevic-Sarkar

The Djodervic-Sarkar4 model is particularly well suited as a broadband model for composite dielectrics. It is defined in terms of the variation of real permittivity ( $\Delta \epsilon$ ), relative high frequency permittivity ( ${\epsilon }_{\infty }$ ), conductivity ( $\sigma$ ), lower limit of angular frequency ( $\omega {}_{1}$ ) and the upper limit of angular frequency ( $\omega {}_{2}$ ).

(6) ${\epsilon }^{*}={\epsilon }_{\infty }+\left[\frac{\Delta \epsilon }{lo{g}_{10}\frac{{\omega }_{2}}{{\omega }_{1}}}\right]\frac{ln\frac{{\omega }_{2}+j\omega }{{\omega }_{1}+j\omega }}{ln\left(10\right)}-\frac{j\sigma }{\omega {\epsilon }_{0}}$

## Frequency List (Linear Interpolation)

Data points at a range of frequencies are specified. Values for the dielectric properties are then linearly interpolated to obtain the dielectric properties at frequency points other than specified. Parameters required are frequency, relative permittivity ( ${\epsilon }_{\text{r}}$ ) and either the loss tangent ( $\mathrm{tan}\delta$ ) or conductivity ( $\sigma$ ).

1 R.Coelho, Physics of dielectrics for the Engineer, 1st ed. Elsevier Scientific Publishing Company, 1979.
2 K.S. Cole and R.H. Cole, “Dispersion and absorption in dielectrics,” Journal of Chemical Physics,vol.9, pp.341-351, 1941
3 J. Baker-Jarvis, M. D. Janezic, J. H. Grosvenor, and R.G. Geyer, “Transmission/reflection and short-circuit line methods for measuring permittivity and permeability: Technical note 1355-r,” National Institute of Standards and Technology, Tech. Rep., 1994
4 Djordjevic, R.M. Biljic, V.D. Likar-Smiljanic, T.K. Sarkar, Wideband frequency-domain characterization of FR4 and time-domain causality, IEEE Transactions. on Electromagnetic Compatibility, vol. 43, no.4, 2001, p.662-667