# The Influence of Capacitance and Inductance Changes on Frequency Response of Transformer Windings
## 3 Model of Inductance and Capacitance Changes on FR Curve
### Modeling Results  
The FEM models have been prepared for the three abovementioned cases: the neutral position is 450 turns, the upper position is 495 turns, and the lower position is 405 turns. In the beginning, the capacitance value in the model was assumed to be \(C=1400\) pF. This value refers to the bushing capacitance, connected parallel to the winding-tank (core) capacitance. Even though the bushing capacitance is dominant, the other one was also evaluated using Maxwell. The analysis of bushing influence of FR curve can be found in the literature [26], where changes are visible in the high frequency range. Authors' experiments show that the FR of a transformer changes in the LF range (up to 10 kHz) after connecting additional capacitance to the bushing in the other phase rather than the tested one, and in this way modifying the total capacitance of the object [9], which will be presented in Section 4. The results of the modeled frequency response are given in Figure 6.  
Figure 5: Complex permeability of the core material (\(\mu_{r}=560\), \(\gamma=2\cdot 10^{6}\) S/m, \(2D=0.23\) mm) as a function of frequency.  
Figure 6: FR modeling for the three positions of the tap changer (405, 450, and 495 turns).  
Although all resulting parameters \(R\), \(L\), and \(C\) differ in the described three models, the inductance changes are dominant for the frequency response at low and middle frequencies. It can be seen that inductance changes influence the whole simulated frequency range, including the first parallel resonance (approximately 2 kHz). FR curve is shifted towards higher frequencies with decrease in number of turns, and toward lower frequencies with increase in the whole frequency range. This effect was expected. In the next step, there the change of bushing capacitance on opposite winding was simulated, which allowed the position of the first resonance resulting from inductance change to be balanced. The results are given in Figure 7.  
It can be seen that the first parallel resonance returned to the original position (nominal tap) after adding capacitance (at 2 kHz). This case may appear in field conditions where there is the wrong tap changer position and capacitance change (e.g., on the bushing insulator of the HV winding) [9]. The change of bushing capacitance can be the result of its earlier replacement or damage, but also by mistakes of testing personnel (leftover wires or tools on bushings of other phases and any other capacitive influence). The interesting behavior can be observed at higher frequencies: from approximately 5 kHz, the curve representing tap position 405, which is wired with additional capacitance, returns to the original position, before adding the capacitance. This means that in the case where the wrong tap position is covered by capacitance changes, this can be easily detected in higher frequencies. On the other hand, changes in FR visible in the MF range may be interpreted as a deformation (because LF range shows no differences), while they can be the result of mentioned \(LC\) phenomena.  
Similarly, this was simulated the contrary case, where turn number was increased, while bushing capacitance decreased (Figure 8).  
Also in this case, the frequency shift of the first parallel resonance may be balanced to the original position by capacitance change. In MF range, the capacitance influence disappears and the curve returns to the original position.  
Figure 7: FR modeling for two positions of the tap changer for 405 and 450 turns, and for 405 turns with bushing capacitance change.