# The Influence of Capacitance and Inductance Changes on Frequency Response of Transformer Windings
## 2 Parameters Influencing the Frequency Response of Transformer Windings  
FRA method, widely used in advanced transformer diagnostics for over a decade, has been standardized [6; 16] and its main problem lies in interpreting detected changes in the shape of FR characteristics. Test results are usually presented as Bode plots, with amplitude values calculated as a signal damping (in dB) and phase angle shift in degrees. The amplitude is often marked as _FRA_, so it can be written:  
\[FRA(dB)=20\log\frac{U_{2}}{U_{1}} \tag{1}\]  
\[\varphi=tg^{-1}(\angle U_{2}-\angle U_{1}) \tag{2}\]  
As was already mentioned, FR can be measured in various test configurations, however, according to the standard [6], the main configuration is end-to-end open test setup, where the signal is applied to the beginning of the winding and measured at its end, with remaining windings left open. All results presented in this paper are based on this configuration, which is presented in Figure 1. For delta connected windings, the signal is given on the bushing connected to the beginning of a tested phase and measured at this point (U\({}_{\text{in}}\)). The second measuring channel is connected to the bushing where the end of the winding in the tested phase is led out (U\({}_{\text{out}}\)), shown in Figure 1a. In the delta winding configuration, part of the signal is also transferred through both remaining phases. In the case of star connected windings, input voltage (U\({}_{\text{in}}\)) is measured on bushings connected to the beginning of the tested phase, while output voltage channel (U\({}_{\text{out}}\)) is connected to the neutral bushing. The signal is transferred only through the tested winding, however obvious couplings to other windings also appear in FR. The frequency response depends also on the column on which the measurement is taken. When measurement results are compared between phases, there are always similar visible differences in the low frequency range. These changes are the effect of various magnetic flux distributions, which for the middle column is symmetrical (via two side columns) and for side columns asymmetrical (via middle one and opposite side columns). Additionally, some smaller differences in the middle frequency range can be observed, if two phases are compared. The FRA results are presented as the frequency function in logarithmic scale, allowing analysis of results in each frequency range, but occasionally linear scale is applied. The FR curve shape is mainly determined by winding inductances and capacitances. During assessment, the FR range is usually divided into three main subranges, sometimes including additional sub-bands. These ranges are as follows: low, medium, and high frequencies. Their exact borders depend on the construction of the transformer, mainly on its geometrical size, usually related to its power rating. Therefore, it is not possible to perform analysis in predefined frequency ranges. The bigger the transformer, the higher the values of inner capacitances, thus the frequency response is shifted into lower frequencies [8]. The behavior at the low frequency (LF) range is usually considered as the interaction of the magnetizing inductance and bulk capacitance to the ground of the winding taken under consideration. In the lowest frequency range (starting at 20 Hz, according to [6]), winding inductances are dominant, resulting in a decreasing characteristic. When the first resonance is reached, the dominant influence of the winding capacitances appears (Figure 2), reaching the end of the LF range. This range is strongly influenced by  
Figure 1: End-to-end open test setup configuration on the example of a Dyn transformer: (**a**) delta winding, (**b**) star winding. short-circuits between the turns or coils, because the magnetic circuit still influences the response in such low frequencies [17; 18; 19].