Complex permittivity (${ϵ}^{\ast }(\omega )={ϵ}^{\prime }(\omega )-i{ϵ}^{\prime \prime }(\omega )$)

For harmonic electric fields with time convention $e^{i\omega t}$, write $\mathbf{D}={ϵ}^{\ast }(\omega )\mathbf{E}$ for their complex amplitudes, where $\omega$ is angular frequency. The real part ${ϵ}^{\prime }$ describes the in-phase electric displacement field; the minus imaginary part ${ϵ}^{\prime \prime }$ describes the loss component. The sign reverses with the opposite time convention. Complex permittivity can also be a tensor in a linear anisotropic dielectric.

With phasor convention $e^{-i\omega t}$, the Ampere-Maxwell law in a scalar conductor reads $\nabla \times \mathbf{H}=(\sigma -i\omega {\epsilon }_0{\epsilon }_r)\mathbf{E}$. Factoring $-i\omega {\epsilon }_0$ gives the displayed effective relative complex permittivity. Its positive imaginary conductive term expresses loss in this convention; the sign reverses for $e^{+i\omega t}$. This combines conduction and displacement currents for the same field and must not be confused with the material's loss-free relative permittivity alone.

Conversion of electrical energy into heat in a dielectric driven by an electric field. In the stated harmonic convention, a passive scalar response has ${ϵ}^{\prime \prime }\ge 0$ for positive angular frequency. A dielectric capacitance measurement measures a dissipative component as well as capacitance; leakage from electrical conductivity can contribute to the inferred loss.

For a capacitor represented by ideal capacitance $C$ and series electrical resistance $R_s$, the complex impedance is $R_s-i/(\omega C)$. The ratio of its dissipative part to the magnitude of its reactive part is $\tan \delta =\omega C{R}_s$. This dimensionless ratio characterizes loss at a specified frequency; equivalent parallel parameters generally differ from the series parameters.

Delayed adjustment of electric polarization after a change in electric field. A simple illustrative model lets the slowly adjusting part ${\mathbf{P}}_s$ obey $\tau d{\mathbf{P}}_s/dt+{\mathbf{P}}_s={ϵ}_0\Delta {ϵ}_r\mathbf{E}$, with a positive response time $\tau$. A harmonic electric field then gives a contribution $\Delta {ϵ}_r/(1+i\omega \tau )$ to complex permittivity divided by vacuum permittivity. This model is an assumption, not a claim that all dielectrics have one response time.