## 2 Overview of the Frequency Dielectric Spectroscopy Measurement and Domain Method
The tangent of angle \(\delta\) is widely referred to as the dissipation factor, defined mathematically as  
\[tg\delta=\frac{\text{Real (resistance) part of Impedance}}{\text{Imaginary (capacitance) part of Impedance}} \tag{1}\]  
In the FDS method, a sinusoidal voltage at a specific frequency is applied to the sample being tested, and the resulting current flowing through the transformer under test is measured. The current through the test object, characterized by its geometric capacitance \(C_{0}\), is described by the equation  
\[I(\omega)=j\omega C_{0}U_{0}\left[\left(\varepsilon_{\infty}+\chi^{\prime}( \omega)\right)-j\left(\frac{\sigma_{dc}}{\varepsilon_{0}\omega}+\chi^{\prime \prime}(\omega)\right)\right] \tag{2}\]  
\[I(\omega)=j\omega C_{0}U_{0}\left[\varepsilon^{\prime}(\omega)-\mathrm{j} \varepsilon^{\prime\prime}(\omega)\right] \tag{3}\]  
where \(C_{0}\) denotes geometric capacitance, \(U_{0}\) represents the applied voltage, \(\sigma_{\text{dc}}\) indicates the DC conductivity, \(\varepsilon_{\infty}\) is the high-frequency permittivity, \(\varepsilon^{\prime}(\omega)\) and \(\varepsilon^{\prime\prime}(\omega)\) are the real and imaginary components of permittivity, respectively, and \(\chi^{\prime}(\omega)\) and \(\chi^{\prime\prime}(\omega)\) represent the real and imaginary sensitivities, respectively [16; 19]. The measured current can be separated into real and imaginary components, corresponding respectively to the in-phase and quadrature-phase currents relative to the applied voltage. The imaginary component of the current represents energy losses, which can be categorized into two main parts:  
* resistive losses associated with conduction currents (primarily due to moisture content),
* dielectric losses resulting from electrical polarization effects [16]. When geometric information about the test object is unavailable, it is most practical to represent the dielectric response using the loss factor (tan \(\delta\)). This factor is expressed as the ratio of the imaginary permittivity \(\varepsilon^{\prime\prime}(\omega)\) to the real permittivity \(\varepsilon^{\prime\prime}(\omega)\), provided by the following equation  
\[tg\delta(\omega)=\frac{\varepsilon^{\prime\prime}(\omega)}{\varepsilon^{ \prime}(\omega)} \tag{4}\]  
It is not possible to clearly identify whether higher energy losses originate primarily from cellulose paper insulation or from insulating oil. Figure 1 illustrates the insulation system of a single-phase column of a distribution transformer. The dielectric response of a transformer's overall insulation system is influenced by several factors, including the characteristics of impregnated paper, pressboard, insulating oil, and the geometric configuration of the insulation components. Precise evaluation of measured data necessitates mathematical analysis, which involves understanding how the properties of both oil and paper/pressboard vary with temperature and moisture content [16]. Figure 2 illustrates the measured frequency dependence of the dissipation factor for pressboard insulation paper containing 1% moisture and oil, highlighting also the effects of interfacial polarization related to insulation geometry. Within the frequency range of 10 to 1000 Hz, cellulose-based insulation typically dominates the dielectric response. The conductivity of transformer oil is influenced by dissolved conductive aging by-products, soot, and high molecular weight acids present in transformer materials, causing fluctuations within this frequency spectrum. The local peak or \"hump\" observed around 0.003 Hz arises from interfacial polarization, which depends significantly on the oil-to-pressboard ratio and the geometric structure of the insulation [16; 20; 21]. Based on the frequency-response analysis presented in Figure 2, a significant limitation of the FDS technique is its sensitivity to temperature variations, which notably impacts the accurate determination of oil conductivity.