# SPICE-Aided Compact Electrothermal Model of Impulse Transformers +
## 2 Model Form
### Non-Linear Thermal Model  
The compact non-linear thermal model, whose network representation is presented in Figure 2, is used to compute temperature of all the transformer components: \(T_{W1}\), \(T_{W2}\), and \(T_{C}\). It takes into account both self-heating in the core, in the primary winding, and in the secondary winding and mutual thermal couplings between each pair of the mentioned components.  
In Figure 2, nine subcircuits are visible. Three of them (shown on the left-hand side) are used to compute three temperatures: \(T_{W1}\), \(T_{W2}\), and \(T_{C}\). They correspond to voltages in  
Figure 2: Network form of a compact non-linear thermal model.  
nodes denoted using the same symbols. In turn, current sources I\({}_{C}\), I\({}_{W1}\), and I\({}_{W2}\) model power dissipated in the core and in each winding.  
The form of this model corresponds to a non-linear thermal model of semiconductor devices proposed in [28]. In the described model, differences in the value of temperature of the core and each winding are taken into account. Additionally, the influence of the dissipated power of the effectiveness of the removal of the heat generated in the transformer is taken into account. While formulating the presented model, it was assumed that the dissipated power influences only thermal resistances \(R_{th}\) in the thermal model, whereas it does not influence thermal capacitances.  
Circuits containing capacitors and the controlled current sources model self-transient thermal impedances: \(Z_{HW1}(t)\) of the primary winding (C\({}_{W11}\),..., C\({}_{W1n}\), G\({}_{W11}\),..., G\({}_{W1n}\)), \(Z_{thW2}(t)\) of the secondary winding (C\({}_{W21}\),..., C\({}_{W2n}\), G\({}_{W21}\),..., G\({}_{W2n}\)), and \(Z_{thC}(t)\) of the core (C\({}_{C1}\),..., C\({}_{Cn}\), G\({}_{C1}\),..., G\({}_{Cn}\)). Voltage on each of these circuits corresponds to an excess of temperature of the transformer components above the ambient temperature \(T_{a}\) due to self-heating. Controlled voltage sources E\({}_{1}\), E\({}_{2}\), and E\({}_{3}\) represent an increase in the temperature of the transformer components resulting from mutual thermal couplings between these components. Voltage on the source E\({}_{1}\) is equal to the sum of voltages in nodes T\({}_{W11}\) and T\({}_{WC1}\), voltage on the source E\({}_{2}\) to the sum of voltages in nodes T\({}_{W21}\) and T\({}_{WC2}\), and voltage on the source E\({}_{3}\) to the sum of voltages in nodes T\({}_{CW1}\) and T\({}_{CW2}\). Voltage sources V\({}_{1}\), V\({}_{2}\), and V\({}_{3}\) correspond to the temperature \(T_{a}\).  
Another six subcircuits describe mutual thermal couplings between the components of the transformer. Each of these subcircuits represents appropriate transfer transient thermal impedance. Current sources represent power dissipated in the primary winding (I\({}_{W12}\) and I\({}_{W1C}\)), the secondary winding (I\({}_{W21}\) and I\({}_{W2C}\)), and the core (I\({}_{C1}\) and I\({}_{C2}\)).  
All self and transfer transient thermal impedances are given by [19]  
\[Z_{th}(t)=R_{th}\cdot\left[1-\sum_{i=1}^{N}a_{i}\cdot\exp\left(-\frac{t}{\tau _{thi}}\right)\right] \tag{12}\]  
The dependence of thermal resistance \(R_{th}\) on power \(p\) dissipated in the heating transformer component is expressed by the empirical formula  
\[R_{th}=R_{th0}\cdot\left[1+a\cdot\exp\left(-\frac{p}{b}\right)\right] \tag{13}\]  
where \(R_{th0}\) is the minimum thermal resistance value, while \(a\) and \(b\) are model parameters. Parameter \(a\) describes the range of change in the thermal resistance value, whereas parameter b characterizes the slope of the dependence \(R_{th}(p)\).  
Thermal resistance is modelled by the controlled current sources \(G_{i}\). The output current of this source is described by  
\[G_{i}=\frac{V_{Gi}}{a_{i}\cdot R_{th}} \tag{14}\]  
where \(V_{Gi}\) is voltage drop on this source.