# SPICE-Aided Compact Electrothermal Model of Impulse Transformers +
## 4 Investigations Results
### Planar Transformers  
Figure 11 illustrates the influence of frequency on the measured and computed values of the core temperature \(T_{C}\) of a planar transformer at two sets of operating conditions characterized by: (a) load resistance \(R_{0}=100\)\(\Omega\) and supply voltage amplitude \(V_{m}=30\) V, (b) \(R_{0}=470\)\(\Omega\) and \(V_{m}=45\) V.  
As can be seen, the temperature \(T_{C}\) decreased with frequency and assumed higher values for the higher of the considered load resistance values. With an increase in \(f\) from 40 to 150 kHz, temperature \(T_{C}\) decreased by 20 \({}^{\circ}\)C. For \(R_{0}=100\)\(\Omega\), the values of temperature \(T_{C}\) computed with the new model and the model from the paper [15] were practically the same, whereas for \(R_{0}=470\)\(\Omega\), these values differed from each other by 15 \({}^{\circ}\)C.  
Figure 12 presents the influence of f on the temperature of the secondary winding \(T_{W2}\) at selected values of \(R_{0}\) and \(V_{m}\).  
The temperature \(T_{W2}\) decreased in function of the frequency. Comparing the values of \(T_{W2}\) and \(T_{C}\) shown in Figures 11 and 12, it can be seen that in the considered operating conditions, the core was warmer than the secondary winding. This means that power of a higher value was dissipated in the core.  
Figures 11 and 12 show that the results of computations obtained using the proposed model were convergent to measurements results for \(f>50\) kHz. At frequency\(f=40\) kHz and lower considered values of load resistance, the computations results were overestimated by 20 \({}^{\circ}\)C. This means that in this range, the proposed model does not correctly describe power dissipated in the transformer components, especi  
Figure 11: Dependences of the temperature \(T_{C}\) on frequency.  
Figure 12: Dependences of temperature of the secondary winding on frequency.  
Figure 13 presents the results of measurements and computations of the transformer energy efficiency \(\eta\) as the function of frequency \(f\) with selected values of load resistance \(R_{0}\).  
As can be seen, the energy efficiency of the transformer increased with an increase in frequency and it decreased if load resistance increased. The efficiency values obtained with resistance \(R_{0}\) equal to 100 \(\Omega\) and 1 k\(\Omega\) differed by three times. A decrease in the efficiency in the range of high load resistance resulted from the high value of no-load current, which became the dominant component of the primary winding current. Increasing the frequency from 40 to 150 kHz increased the energy efficiency by up to 15%.  
Figure 14 presents the non-isothermal electrical characteristic of the planar core power supplied by direct current. In this case, in the computer analysis, the core was connected to the supplied networks using connectors X\({}_{1}\) and X\({}_{2}\) of the proposed model.  
As can be observed, the obtained dependence \(v(i)\) possessed the maximum at current \(i=50\) mA. The obtained shape of the considered characteristic was a result of a self-heating phenomenon and an increasing function of the core resistance on temperature \(T_{C}\)[25].