# SPICE-Aided Compact Electrothermal Model of Impulse Transformers +
## 4 Investigations Results
### Ring Transformers  
Figure 5 illustrates the influence of load resistance \(R_{0}\) on energy efficiency \(\eta\) of the considered ring transformers. This efficiency was equal to the quotient of the average value of the product of voltage on the secondary winding \(V_{W2}\) and the current of this winding I\({}_{W2}\) by the average value of the product of the voltage on the primary winding \(V_{W1}\) and the current of this winding I\({}_{W1}\). The obtained characteristics \(\eta(R_{0})\) were decreasing functions for all the transformers. The highest energy efficiency was obtained for the transformer containing the nanocrystalline core (RTN), whereas the lowest was for the transformer containing the core made of powdered iron (RTP). At the mentioned operating conditions, for the transformer with the RTP core, the values of \(\eta\) decreased over 14 times (from 85% to only just 6%). A decrease in  
Figure 4: Investigated transformer with ring cores: (**a**) RTN, (**b**) RTP, and (**c**) RTF. Figure 3: Investigated planar transformer with an E22/6/16R core made of 3F3 material. energy efficiency for high values of \(R_{0}\) resulted from a high value of idling current, which is the main component of current flowing through the primary winding. As one can notice, the computations results obtained using the proposed model were convergent with the measurements results for all the considered transformers in the whole range of \(R_{0}\) changes. In contrast, the results obtained using the model from [15] fit well the measurements results only for the RTP core, whereas they differed from the results of measurements even by 15% for the other transformers. Figure 6 illustrates an influence of load resistance on the core temperature at \(V_{m}=67\) V and \(f=100\) kHz. As seen, the dependence \(T_{C}(R_{0})\) was an increasing function. The core temperature achieved even 110 \({}^{\circ}\)C for the core RTP at \(R_{0}=1\) k\(\Omega\). The obtained shape of the characteristics \(T_{C}(R_{0})\) shows that the most important component of transformer power losses was dissipated in the core [27]. The values of the temperature T\({}_{C}\) computed using the new model fit the measurements results well. Differences between them were lower than 5 \({}^{\circ}\)C. In contrast, the results of computations obtained using the model from the paper [15] differed visibly from the results of measurements. Such differences reached as far as 50 \({}^{\circ}\)C.  
Figure 7 illustrates the dependence of the primary winding temperature \(T_{W1}\) on load resistance at frequency \(f=100\) kHz. Figure 5: Dependences of energy efficiency of the ring transformers on load resistance. Figure 6: Dependences of the core temperature on load resistance. For the transformers with RTP and RTF cores, the dependence \(T_{w1}(R_{0})\) was an increasing function, whereas for the transformer with RTN core, the local minimum at \(R_{0}=100\)\(\Omega\) could be observed. Big differences (equal even to 70 \({}^{\circ}\)C) in the values of temperature \(T_{W1}\) for different transformers were observed at high values of load resistance. This means that this value was a result of mutual thermal couplings between the primary winding and the core, because in this range of \(R_{0}\), the power was dissipated mostly in the core. The results obtained with the proposed model fit the results of measurements well only for the transformer with the RTP core in the range of high \(R_{0}\) values. The differences between the measurements and computations results did not exceed 3 \({}^{\circ}\)C. In contrast, the results obtained using the model given in [15] differed from the measurements results by 15 \({}^{\circ}\)C.  
Comparing results presented in Figures 6 and 7, one can observe that using the model described in the paper [15], one can obtain overestimated values of the transformer core and underestimated values of the transformer primary winding.