Transformer Model

Transformer Excitation

Theoretical Background

In the time domain, the transformer excitation voltage~$e(t)$\,[V] and magnetization current~$i^m(t)$\,[A] are related through the BH-curve:

\[\begin{align} e(t) \rightarrow B(t) \stackrel{\mbox{BH-curve}}{\rightarrow} H(t) \rightarrow i^m(t) \end{align}\]

Illustration of the relationship between the excitation voltage~$e(t)$ and magnetization current~$i^m(t)$

The transformer excitation voltage~$e(t)$ relates to its magnetic flux density~$B(t)$\,[T]:

\[\begin{align} B(t) &= \sum_{h \in H} \frac{|E_h|}{A \cdot \omega_h} \cdot \cos(\omega_h \cdot t + \theta_h), \\ \mbox{given:} & \nonumber \\ e(t) &= \sum_{h \in H} |E_h| \cdot \sin(\omega_h \cdot t + \theta_h), \\ B(t) &= \frac{1}{A} \int -e(t) \mathrm{d}t, \end{align}\]

where~$A$\,[m^2], $\omega_h$\,[rad/Hz], $|E_h|$\,[V] and~$\theta_h$\,[rad] denotes the core surface, harmonic angular frequency, harmonic excitation voltage magnitude and phase angle, respectively. The link between the frequency-domain excitation current~$E$\,[pu] and the time-domain excitation voltage~$e(t)$, and consequently the magnetic flux density~$B(t)$, is provided through the following functions, depending on the chosen excitation voltage formulation E_formulation, i.e., :polar or :rectangular.

The transformer magnetization current~$i^m(t)$ relates to its magnetic field intensity~$H(t)$\,[Ampere-turn/meter]:

\[\begin{align} i^m(t) &= H(t) \cdot l, \end{align}\]

where~$l$\,[m] denotes the mean magnetic path. The frequency-domain magnetization current~$I^{e}$\,[pu] is determined through a Fourrier transform of the time-domain magnetization current~$i^m(t)$, adjusted for current basis. Depending on the chosen excitation voltage formulation E_formulation, the frequency-domain magnetization current is expressed in :polar or :rectangular coordinates.

Implementation

All excitation data are stored in a dictionary xfmr_exc with:

• key = id of the xfmr [Int]
• val = a dictionary [Dict{String,Any}] consisting of the following input:
  • General input, including: