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# Frequency Domain Measurement

The purpose is to measure the complex permittivity spectra ε*(ω) and conductivity spectra σ*(ω) of a sample material. The material may be solid or liquid and is usually placed between two electrodes which form a capacitor.

For compact formulas, complex notation is used and indicated by the asterisk.

From the sample capacitors impedance spectrum

$Z^*(\omega) = Z'(\omega)+jZ''(\omega)=\frac{U^*(\omega)}{I^*(\omega)}$

the complex sample capacity

$C^*(ω) :=\dfrac{1}{jωZ^*(ω)}$

is calculated. C*(ω) is directly related to the permittivity spectrum

$\epsilon^*(\omega)=\epsilon'-j\epsilon''=\frac{C^*(\omega)}{C_0}$

where C0 is the capacity of the empty electrodes without sample material in between. If the electrodes are arranged parallel

$C_0=\epsilon_0\frac Ad$

where A denotes the area of one electrode, d the spacing between the electrodes and ε0=8.854*10-12 As/Vm is the vacuum permittivity. It should be noted that ε*(ω) generally includes a permittivity contribution from molecular dipoles and a conductivity contribution from free charge carriers. The same applies to the conductivity spectrum which is another representation of ε*(ω), i.e.,

$\sigma^*(\omega)=j\omega\epsilon_0(\epsilon^*(\omega)-1)$

The term −1 takes the charge contribution of the empty electrodes (vacuum permittivity)  into account, resulting in ε*(ω) = 1 and σ*(ω) = 0 for the vacuum. Refer to Dielectric Spectroscopy and Impedance Spectroscopy of Materials for details and typical properties of ε*(ω) and σ*(ω).