# SPICE-Aided Compact Electrothermal Model of Impulse Transformers +
## 1 Introduction  
Impulse transformers are commonly used in many applications, for example in switch-mode power converters [1, 2, 3, 4, 5]. Such converters require considered components operating at frequency ranging from several to several hundred kilohertz [4, 6]. The considered transformers contain two components: a core made of any ferromagnetic material and windings (at least two) made of conductive material [1, 3, 4]. During the operation of a transformer, different phenomena, magnetic, electrical, and thermal, occur in it. Magnetic phenomena in the core are a result of the excitation of magnetic force connected with the current flow through the transformer windings. On the other hand, alternating magnetic force induces electromotive force (induced voltage) in all the transformer windings. Additionally, temperature influences electrical properties of the windings and magnetic properties of the ferromagnetic core [4, 6, 7, 8]. Due to thermal phenomena occurring in the transformer, temperatures of the core and the windings can be much higher than the ambient temperature. These phenomena contain self-heating in each transformer components and mutual thermal couplings between these components [9]. Currently, engineers use computer programs, e.g., SPICE or PLECS, to design and analyze power electronic circuits [10, 11, 12]. Practical usefulness of the obtained results of computations depends on, e.g., accuracy of the models of applied elements contained in these circuits [13]. Therefore, models of the considered components characterized by a different accuracy are given in the literature. The review of selected transformers models isgiven in the papers [14; 15]. Additionally, a manner of modeling the thermal properties of selected impulse transformers is described in the papers [6; 9; 16]. From the literature review, it is visible that the accessible transformers models describe properties of this component in a too-simplified way. This means that such models can correctly describe properties of impulse transformers only at selected operating conditions. The very popular linear model of the ideal transformer containing linearly coupled inductors [1; 10; 17] is accurate only in the case when the magnetic permeability of the transformer core is constant. This condition is fulfilled when the dependence of magnetic flux density \(B\) on magnetic force \(H\) is linear. Such dependence \(B(H)\) is observed when \(B\) value is considerably lower than saturation flux density. The mentioned simple model neither takes into account thermal phenomena occurring in the transformer nor power losses in the core and in the windings. On the other hand, non-linear transformer models given in [17] make it possible to take into account non-linearity of the magnetization characteristics \(B(H)\) of the ferromagnetic core. Unfortunately, in these models the influence of temperature on transformer characteristics are omitted. In order to take into account mutual interactions between magnetic and thermal phenomena in computer analyses, the electrothermal models of the components contained in the investigated circuit are needed. For transformers, inductors, or solid-state light sources, such models belong to multi-domain models [18]. Electrothermal models can be formulated in the form of detailed or compact models [19; 20]. The detailed models allow computing time-spatial temperature distribution in the modeled device, but computations with the use of such models are time-consuming. In turn, compact models make it possible to compute a waveform of one internal temperature characterizing the whole device and computations using such models in a short time. Electrothermal transformer models are given in the literature [7; 14]. Models used in these papers are based on the ferromagnetic core model proposed in [21] and extended by the equations describing the temperature influence on the dependence \(B(H)\).