# The Influence of Capacitance and Inductance Changes on Frequency Response of Transformer Windings
## 3 Model of Inductance and Capacitance Changes on FR Curve
### Core Modeling  
Proper representation of ferromagnetic core parameters is required for an accurate simulation of frequency response of the winding in the wide frequency range. The ferromagnetic core has significant influence on the frequency response of the winding below approximately 10 kHz, therefore correct simulation of the core material parameters below 10 kHz is especially important. The behavior of the core material is modeled using complex permeability. Equations (11)-(15) were used to evaluate the magnetic permeability given for Maxwell for each frequency. Eddy currents flow in the r-z plane, so it is possible to model it with only one component of magnetic vector potential \(\mathbf{A}_{\phi}\). The model is based on an 800 kVA transformer, which is an old unit, and its core is made from cold-rolled, non-grain-oriented steel, which was assumed in the model. Assuming one-dimensional propagation of the electromagnetic wave into the laminated ferromagnetic core with 2D sheet thickness, the wave equation can be written as [24]:  
\[H_{y}(z)=H_{y0}\frac{\cosh kz}{\cosh kD} \tag{11}\]  
where \(k=\sqrt{\
icefrac{{\mu\
u\mu}}{{\gamma}}}=\frac{(1+\
icefrac{{\mu}}{{\beta}})}{\beta}\) and \(\delta\) is the skin-depth \(\delta=\sqrt{\frac{2}{\omega\mu_{T}}}\).  
Complex permeability in laminated core is given by the equation [25]:  
\[\underline{\mu}=\frac{1}{H_{y0}}\cdot\overline{B}=\frac{1}{H_{y0}\cdot 2D} \int\limits_{-D}^{D}\left(\mu H_{y}\right)dz=\mu_{0}\mu_{r}\cdot\frac{\sinh kD }{kD\cdot\cosh kD} \tag{12}\]  
where \(\underline{\mu}\) is complex permeability, \(H_{y0}\) is magnetic field intensity on the surface, \(\overline{B}\) is average magnetic flux density, \(\mu_{0}\) is vacuum permeability, and \(\mu_{r}\) is relative permeability, which can be split into real and imaginary parts:  
\[\underline{\mu}=\mu_{0}\left(\mu^{\prime}+j\mu^{\prime\prime}\right) \tag{13}\]  
Real and imaginary parts take the form of:  
\[\mu^{\prime}=\frac{\text{Re}\left(\underline{\mu}\right)}{\mu_{0}}=\frac{\mu_ {r}\delta}{2D}\left(\frac{\sinh\left(\frac{2D}{\delta}\right)+\sin\left(\frac{ 2D}{\delta}\right)}{\cosh\left(\frac{2D}{\delta}\right)+\cos\left(\frac{2D}{ \delta}\right)}\right) \tag{14}\]  
\[\mu^{\prime\prime}=\frac{-\text{Im}\left(\underline{\mu}\right)}{\mu_{0}}= \frac{\mu_{r}\delta}{2D}\left(\frac{\sinh\left(\frac{2D}{\delta}\right)-\sin \left(\frac{2D}{\delta}\right)}{\cosh\left(\frac{2D}{\delta}\right)+\cos \left(\frac{2D}{\delta}\right)}\right) \tag{15}\]  
Figure 5 presents complex permeability as a function of frequency. The real part of the complex permeability represents the ability of the core material to conduct the magnetic flux, while the imaginary part represents the core losses caused by eddy currents circulating inside the laminations.  
Figure 4: Network used for calculations of FRA characteristics. The circuit in the bottom relates to the bushing.  
The complex permeability is calculated taking into account the maximum equivalent permeability of the ferromagnetic material, equivalent anisotropic conductivity, and the thickness of the steel sheet. In the computer analysis, it is assumed that the permeability of the material varies with the frequency of the real part of the complex permeability. The values of magnetic permeability shown in Figure 5 were introduced in Maxwell input data for each frequency separately.