# SPICE-Aided Compact Electrothermal Model of Impulse Transformers +
## 2 Model Form
### Core Model  
In the core model, the voltage on terminal B corresponds to magnetic flux density, the voltage on terminal H to magnetic force, and the voltage on terminal Ploss to power dissipated in the core, whereas terminals X1 and X2 represent the border of the core. The controlled voltage source E4 is used to compute magnetization on the primary magnetizing curve \(M_{a}\) given by [26]  
\[M_{a}=\left[\frac{B_{S0}\cdot(1+\alpha_{BS}\cdot(T_{C}-T_{0}))}{\mu_{0}}-H_{S} \right]\cdot f\left(\frac{H+\alpha_{M}\cdot M_{a}}{A}\right) \tag{1}\]  
where \(B_{S0}\) is saturation flux density at the reference temperature \(T_{0}\) and magnetic force \(H_{S}\), \(\alpha_{BS}\)--the temperature coefficient of saturation flux density, \(\mu_{0}\)--magnetic permeability of free air, \(T_{C}\)--core temperature, \(\alpha_{M}\)--the parameter of coupling of the walls of magnetic domains, and \(A\)--the parameter of thermal energy. All the abovementioned parameters also appear in the Jiles-Atherton model. In this model, _f(x)_ is the Langevin function, which is non-continuous for \(x=0\). In order to eliminate this disadvantage, in the proposed model, this function is described by the empirical formula [26]  
\[f(x)=\frac{x}{\sqrt{x^{2}+1}}\left[1-0.9\cdot\exp(-\frac{|x|}{2.5})-0.1\cdot \exp(-\frac{|x|}{25})\right] \tag{2}\]  
Voltage drop on resistor R3 is proportional to time derivative \(dM_{a}/dt\). Controlled voltage source E5 makes it possible to compute flux density with the use of the following formula  
\[B=\mu_{0}\cdot(H+M\cdot C_{R}\cdot y) \tag{3}\]  
where \(C_{R}\) denotes capacitance of capacitor of the same name, \(y\) describes the Curie phenomenon with the empirical formula  
\[y=\left\{ \begin{array}{ll}1\;\;if\;T_{C}<T_{Curie}\\ 1-(T_{C}-T_{Curie})\cdot 0.006K^{-1}\;\;if\;T_{Curie}<T_{C}<T_{[PERSON]}+17K\\ 0\;\;if\;T_{C}>T_{[PERSON]}+17K\end{array} \right. \tag{4}\]  
and \(T_{Curie}\) denotes the Curie temperature. In turn, voltage on resistor R2 is proportional to time derivative \(dB/dt\). Controlled voltage source E\({}_{\rm H}\) represents magnetic force \(H\) and voltage drop on resistor R1 is proportional to \(dH/dt\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name of Variable** & **Symbol** & **SPICE Voltage** \\ \hline Magnetic force & \(H\) & v(H) \\ \hline Magnetic flux density & \(B\) & v(B) \\ \hline Magnetization & \(M\) & v(M) \\ \hline Power losses in the core & \(P_{\rm loss}\) & v(Ploss) \\ \hline Voltage between the core terminals & \(v\) & v(X1,X2) \\ \hline Temperature of the core & \(T_{C}\) & V(TC) \\ \hline Temperature of the primary winding & \(T_{\rm W1}\) & V(TW1) \\ \hline Temperature of the secondary winding & \(T_{\rm W2}\) & V(TW2) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relations between thermal and magnetic variables and SPICE voltages. Voltage in the node \(M\) represents magnetization of the core. The output current of the controlled current source \(G_{1}\) is given by [26]  
\[G_{1}=-\frac{(M_{a}-M\cdot C_{1})\cdot\text{sgn}(dH/dt)}{C_{1}\cdot H_{C0}\cdot [1+\alpha_{HC}\cdot(T_{C}-T_{0})]}\cdot\frac{dH}{dt}-\frac{C}{C_{1}\cdot(1+C) }\cdot\frac{dM_{a}}{dt}+M\cdot\frac{C_{R}}{R_{C}} \tag{5}\]  
where function _sgn(z)_ computes the sign of the argument \(z\), \(H_{C0}\)--coercion magnetic force at the reference temperature \(T_{0}\), \(C\) characterizes elastic deformations of domains walls, \(\alpha_{HC}\)--the temperature coefficient of coercion magnetic force, and \(C_{1}\) and \(R_{C}\) are capacitance of the capacitor and resistance of the resistor occurring in the core model. Controlled voltage sources E\({}_{\text{alf}}\), E\({}_{C}\), E\({}_{\text{u}}\), and E\({}_{\text{A1}}\) are used to compute the values of the parameters \(\alpha\), \(C\), \(\mu\), and \(A\) occurring in equations describing the \(B(H)\) curve. The dependences of each of the mentioned parameters on temperature \(T_{C}\) are given by linear functions.