## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## Complex permittivi

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field. The response must always be causal (arising after the applied field) which can be represented by a phase difference. For this reason permittivity is often treated as a complex function (since complex numbers allow specification of magnitude and phase) of the (angular) frequency of the applied field ω, $varepsilon ightarrow widehat{varepsilon}(omega)$. The definition of permittivity therefore becomes

$D_0 e^{-i omega t} = widehat{varepsilon}(omega) E_0 e^{-i omega t},$

where

D0 and E0 are the amplitudes of the displacement and electrical fields, respectively,
i is the imaginary unit, i 2 = −1.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity εs (also εDC ):

$varepsilon_{ ext{s}} = lim_{omega ightarrow 0} widehat{varepsilon}(omega).$

At the high-frequency limit, the complex permittivity is commonly referred to as ε. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase difference δ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength (E0), D and E remain proportional, and

$widehat{varepsilon} = frac{D_0}{E_0} = |varepsilon|e^{idelta}.$

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:

$widehat{varepsilon}(omega) = varepsilon'(omega) + ivarepsilon''(omega) = frac{D_0}{E_0} left( cosdelta + isindelta ight).$

where

ε′ is the real part of the permittivity, which is related to the stored energy within the medium;
ε″ is the imaginary part of the permittivity, which is related to the dissipation (or loss) of energy within the medium.

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