Debye relaxation

The Debye relaxation (after Peter Debye ) describes temporal reversal processes of the electrical polarization of a material with the help of first-order attenuators . So it is a special case of the supercritically damped dielectric resonance .

Mathematical description

Dielectric resonant circuit

The static dielectric constant of the material is defined from the relationship between the electrical flux density and the electrical field strength in the material . In the Dieletrizität is with designated. The following applies: ${\displaystyle D}$ ${\displaystyle E_{m}}$ ${\displaystyle \omega =0}$ ${\displaystyle \varepsilon _{s}}$ ${\displaystyle \omega =\infty }$ ${\displaystyle \varepsilon _{\infty }}$

${\displaystyle D(t)=\varepsilon _{0}\varepsilon _{s}E_{m}(t)}$,

${\displaystyle \tau {\dot {D}}(t)=\varepsilon _{0}\varepsilon _{\infty }\tau {\dot {E}}_{m}(t)}$,

${\displaystyle {\frac {1}{\omega _{0}^{2}}}{\ddot {D}}(t)=\varepsilon _{0}\varepsilon _{\infty }{\frac {1}{\omega _{0}^{2}}}{\ddot {E}}_{m}(t)}$

${\displaystyle \omega _{0}}$is the resonance frequency , is the relaxation time. Adding all three equations gives: ${\displaystyle \tau }$

${\displaystyle {\frac {1}{\omega _{0}^{2}}}{\ddot {D}}(t)+\tau {\dot {D}}(t)+D(t)=\varepsilon _{0}\varepsilon _{\infty }{\frac {1}{\omega _{0}^{2}}}{\ddot {E}}_{m}(t)+\varepsilon _{0}\varepsilon _{\infty }\tau {\dot {E}}_{m}(t)+\varepsilon _{0}\varepsilon _{s}E_{m}(t)}$

The Fourier transform of the equation is therefore:

${\displaystyle \left(-{\frac {\omega ^{2}}{\omega _{0}^{2}}}+j\omega \tau +1\right)D(j\omega )=\varepsilon _{0}\left(-\varepsilon _{\infty }{\frac {\omega ^{2}}{\omega _{0}^{2}}}+j\omega \varepsilon _{\infty }\tau +\varepsilon _{s}\right)E_{m}(j\omega )}$

The following applies to the complex frequency-dependent electrical permittivity : ${\displaystyle {\underline {\varepsilon }}_{r}}$

${\displaystyle {\underline {\varepsilon }}_{r}=\varepsilon _{r}'-j\varepsilon _{r}''={\frac {D(j\omega )}{\varepsilon _{0}E_{m}(j\omega )}}={\frac {-\varepsilon _{\infty }{\frac {\omega ^{2}}{\omega _{0}^{2}}}+j\omega \varepsilon _{\infty }\tau +\varepsilon _{s}}{-{\frac {\omega ^{2}}{\omega _{0}^{2}}}+j\omega \tau +1}}={\frac {\varepsilon _{s}-\varepsilon _{\infty }}{-{\frac {\omega ^{2}}{\omega _{0}^{2}}}+j\omega \tau +1}}+\varepsilon _{\infty }}$

The division into the real part and the imaginary part now results in: ${\displaystyle \varepsilon _{r}'}$ ${\displaystyle \varepsilon _{r}''}$

${\displaystyle \varepsilon _{r}'={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)\left(1-{\frac {\omega ^{2}}{\omega _{0}^{2}}}\right)}{\left(1-{\frac {\omega ^{2}}{\omega _{0}^{2}}}\right)^{2}+\left(\omega \tau \right)^{2}}}+\varepsilon _{\infty }}$,

${\displaystyle \varepsilon _{r}''={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)\omega \tau }{\left(1-{\frac {\omega ^{2}}{\omega _{0}^{2}}}\right)^{2}+\left(\omega \tau \right)^{2}}}}$

Borderline case for Debye relaxation

If the relaxation times are much longer than the inverse resonance frequency , the material is subject to Debye relaxation. The second order derivatives can therefore be neglected and the following applies: ${\displaystyle \tau \gg \omega _{0}^{-1}}$

${\displaystyle \varepsilon _{r}'={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)}{1+\left(\omega \tau \right)^{2}}}+\varepsilon _{\infty }}$

${\displaystyle \varepsilon _{r}''={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)\omega \tau }{1+\left(\omega \tau \right)^{2}}}}$

The plot of against is called the Cole-Cole diagram , the direct relationship between and is called the Kramers-Kronig relation and the loss factor is defined as: ${\displaystyle \varepsilon _{r}'}$ ${\displaystyle \varepsilon _{r}''}$ ${\displaystyle \varepsilon _{r}'}$ ${\displaystyle \varepsilon _{r}''}$

${\displaystyle \tan \left(\delta \right)={\frac {\varepsilon _{r}''}{\varepsilon _{r}'}}}$

The following applies to the loss factor maximized over the frequency :

${\displaystyle \tan \left(\delta \right)_{\mathrm {max} }={\frac {\varepsilon _{s}-\varepsilon _{\infty }}{2{\sqrt {\varepsilon _{s}\varepsilon _{\infty }}}}}}$

literature

• Ellen Ivers-Tiffee, Waldemar von Münch: Materials in electrical engineering. 10th edition. Teubner Verlag, 2007, ISBN 978-3-8351-0052-7 .