# The Influence of Capacitance and Inductance Changes on Frequency Response of Transformer Windings
## 3 Model of Inductance and Capacitance Changes on FR Curve
### 2D Computer Model of the Winding in FEM Software  
The model is based on the Finite Elements Method (FEM) in two-dimensional cylindrical symmetry. The geometry of the model shown in Figure 3 corresponds to the dimensions of the typical LV winding of the distribution transformer (800 kVA, 15/0.4 kV, Dyn5). The complicated construction of the actual transformer makes the 3D simulation computationally costly and impractical, thus it was necessary to use the simplified 2D model. During the analysis, the solver assumed that the 2D geometry of the model being studied sweeps 360\({}^{\circ}\) around the z-axis of a cylindrical coordinate system. This model, in particular, does not take into account the existence of the other core columns, but it allows for the simulation to be performed, showing the influence of \(C\) and \(L\) parameters with satisfactory accuracy. Simulated cross-over winding nominally has 450 turns. Separate models were made to show the influence of changing the position of the tap changer +/- 10% in tapping turn range. These models are similar to the basic one shown in Figure 3, with the difference being that they contain 405 and 495 turns. One section of the winding has been magnified to show that all turns with inter-wire insulation were taken into account during analysis. On the top of the core window, the additional wire provides the connection to the bushing capacitance. This influence of bushings can be modeled as a single element, because during the measurement the winding is seen as a concentrated inductance and capacitance. The treatment as additional wire also allows modification of bushing capacitances [21; 22]. On external boundaries, the so-called \"ballooning\" boundary condition has been stated. It allows one to cut off the region of analysis near to the object. The software used for the simulations was ANSYS Maxwell v. 19.2, especially the 2D eddy current solver, which computes steady-state, time-varying (AC) magnetic fields at a given frequency. The following equations (3)-(10) were chosen in the Maxwell calculation package. They allow calculation of necessary \(R\), \(L\), and \(C\) parameters. The eddy current field solver calculates the electromagnetic field by solving the magnetic vector potential and the electric scalar potential in the field equation:  
\[\
abla\times\frac{(\
abla\times\mathbf{A})}{\mu}=(\gamma+j\omega\varepsilon)( -j\omega\mathbf{A}-\
abla\phi) \tag{3}\]  
Figure 3: 2D FEM model of 450 turn LV winding. where **A** is magnetic vector potential, \(\phi\) is electric scalar potential, \(\mu\) is magnetic permeability, \(\omega\) is angular frequency at which all quantities are oscillating, \(\gamma\) is conductivity, and \(\varepsilon\) is permittivity. The inductances of the coil, both self and mutual, have been computed by ANSYS Maxwell using the energy of the electromagnetic field delivered by FEM:  
\[W_{AV}=\frac{1}{8}\int\textbf{B}\cdot\textbf{H}^{*}d\Omega \tag{4}\]  
\[L=\frac{8W_{AV}}{I^{2}{}_{Peak}}=\int\textbf{B}\cdot\textbf{H}^{*}d\Omega \tag{5}\]  
where \(W_{AV}\) is average energy of the magnetic field, \(L\) is inductance, **B** is magnetic flux density, \(\textbf{H}^{*}\) is magnetic field intensity conjugate, and \(I_{Peak}\) is peak current value. Wire resistance values were calculated from the power losses \(P\) due to current flow with density \(J\):  
\[P=\frac{1}{2\gamma}\int\textbf{J}\cdot\textbf{J}^{*}d\Omega \tag{6}\]  
\[R=\frac{2P}{I^{2}_{Peak}}=\frac{\int\textbf{J}\cdot\textbf{J}^{*}d\Omega}{ \gamma I^{2}_{Peak}} \tag{7}\]  
In order to calculate self and mutual capacitances, the following electrostatic field equation has been solved:  
\[\
abla\cdot(\varepsilon_{r}\varepsilon_{0}\
abla\phi(r,z))=-\rho \tag{8}\]  
where \(\rho\) is charge density, \(\phi\) is scalar potential of the electric field induced with charge density \(\rho\), \(\varepsilon_{0}\) is vacuum permeability, and \(\varepsilon_{r}\) is relative permeability of the environment.