linguistic:voice_change
Frequency-dependent capacitors

If a capacitor is driven with a time-varying voltage that changes rapidly enough, at some frequency the polarization of the dielectric cannot follow the voltage. As an example of the origin of this mechanism, the internal microscopic dipoles contributing to the dielectric constant cannot move instantly, and so as frequency of an applied alternating voltage increases, the dipole response is limited and the dielectric constant diminishes. A changing dielectric constant with frequency is referred to as dielectric dispersion, and is governed by dielectric relaxation processes, such as Debye relaxation. Under transient conditions, the displacement field can be expressed as (see electric susceptibility): D (t) = ε 0 ∫ − ∞ t ε r (t − t ′) E (t ′) d t ′, {\displaystyle {\boldsymbol {D(t)}}=\varepsilon _{0}\int _{-\infty }^{t}\varepsilon _{r}(t-t'){\boldsymbol {E}}(t')\,dt',}

indicating the lag in response by the time dependence of ε, calculated in principle from an underlying microscopic analysis, for example, of the dipole behavior in the dielectric. See, for example, linear response function. The integral extends over the entire past history up to the present time. A Fourier transform in time then results in: D (ω) = ε 0 ε r (ω) E (ω), {\displaystyle {\boldsymbol {D}}(\omega)=\varepsilon _{0}\varepsilon _{r}(\omega){\boldsymbol {E}}(\omega)\,,}

where ε (ω) is now a complex function, with an imaginary part related to absorption of energy from the field by the medium. See permittivity. The capacitance, being proportional to the dielectric constant, also exhibits this frequency behavior. Fourier transforming Gauss's law with this form for displacement field:

I (ω) = j ω Q (ω) = j ω ∮ Σ D (r, ω) ⋅ d Σ = V (ω) = V (ω) Z (ω), {\displaystyle {\begin{aligned}I(\omega)&=j\omega Q(\omega)=j\omega \oint _{\Sigma }{\boldsymbol {D}}({\boldsymbol {r}},\omega)\cdot d{\boldsymbol {\Sigma }}\\&=\leftV(\omega)={\frac {V(\omega)}{Z(\omega)}}\,,\end{aligned}}} where j is the imaginary unit, V (ω) is the voltage component at angular frequency ω, G (ω) is the real part of the current, called the conductance, and C (ω) determines the imaginary part of the current and is the capacitance. Z (ω) is the complex impedance.

When a parallel-plate capacitor is filled with a dielectric, the measurement of dielectric properties of the medium is based upon the relation: ε r (ω) = ε r ′ (ω) − j ε r ″ (ω) = 1 j ω Z (ω) C 0 = C cmplx (ω) C 0, {\displaystyle \varepsilon _{r}(\omega)=\varepsilon '_{r}(\omega)-j\varepsilon ''_{r}(\omega)={\frac {1}{j\omega Z(\omega)C_{0}}}={\frac {C_{\text{cmplx}}(\omega)}{C_{0}}}\,,} where a single prime denotes the real part and a double prime the imaginary part, Z (ω) is the complex impedance with the dielectric present, C (ω) is the so-called complex capacitance with the dielectric present, and C is the capacitance without the dielectric. (Measurement "without the dielectric" in principle means measurement in free space, an unattainable goal inasmuch as even the quantum vacuum is predicted to exhibit nonideal behavior, such as dichroism. For practical purposes, when measurement errors are taken into account, often a measurement in terrestrial vacuum, or simply a calculation of C, is sufficiently accurate.)

Using this measurement method, the dielectric constant may exhibit a resonance at certain frequencies corresponding to characteristic response frequencies (excitation energies) of contributors to the dielectric constant. These resonances are the basis for a number of experimental techniques for detecting defects. The conductance method measures absorption as a function of frequency. Alternatively, the time response of the capacitance can be used directly, as in deep-level transient spectroscopy.

Another example of frequency dependent capacitance occurs with MOS capacitors, where the slow generation of minority carriers means that at high frequencies the capacitance measures only the majority carrier response, while at low frequencies both types of carrier respond.

At optical frequencies, in semiconductors the dielectric constant exhibits structure related to the band structure of the solid. Sophisticated modulation spectroscopy measurement methods based upon modulating the crystal structure by pressure or by other stresses and observing the related changes in absorption or reflection of light have advanced our knowledge of these materials.