Question 7.8: NEARLY DEBYE RELAXATION There are some dielectric solids tha...

The loss peak occurs when $\omega_{o}=1 / \tau,  \text { so that } \tau=1 / \omega_{o}=1 /(2 \pi 6000)=26.5 \mu s$. We can now calculate the real and imaginary parts of $\varepsilon _{r}$ at 29 kHz,

$\varepsilon_{r}^{\prime}=\varepsilon_{r \infty}+\frac{\varepsilon_{r d c}-\varepsilon_{r \infty}}{1+(\omega \tau)^{2}}=2.58+\frac{3.6-2.58}{1+\left[(2 \pi)\left(29 \times 10^{3}\right)\left(26.5 \times 10^{-6}\right)\right]^{2}}=2.62$

$\varepsilon_{r}^{\prime \prime}=\frac{\left(\varepsilon_{r dc }-\varepsilon_{r \infty 0}\right)(\omega \tau)}{1+(\omega \tau)^{2}}=\frac{(3.6-2.58)\left[(2 \pi)\left(29 \times 10^{3}\right)\left(26.5 \times 10^{-6}\right)\right]}{1+\left[(2 \pi)\left(29 \times 10^{3}\right)\left(26.5 \times 10^{-6}\right)\right]^{2}}=0.202$

and hence

$\tan \delta=\frac{\varepsilon_{r}^{\prime \prime}}{\varepsilon_{r}^{\prime}}=\frac{0.202}{2.62}=0.077$

which is close to the experimental value of 0.084.

This example was a special case of nearly Debye relaxation. Debye equations have been modified over the years to account for the broad relaxation peaks that have been observed, particularly in polymers, by writing the complex $\varepsilon _{r}$ as

$\varepsilon_{r}=\varepsilon_{r \infty}+\frac{\varepsilon_{r dc }-\varepsilon_{r \infty}}{\left[1+(j \omega \tau)^{\alpha}\right]^{\beta}}$             [7.36]

where α and β are constants, typically less than unity (setting α = β = 1 generates the Debye equations). Such equations are useful in engineering for predicting $\varepsilon _{r}$ at any frequency from a few known values at various frequencies, as highlighted in this simple nearly Debye example. Further, if $\tau$ dependence on the temperature $T$ is known (often $\tau$ is thermally activated), then we can predict $\varepsilon _{r}$ at any $\omega$ and $T$.