1. Application of the Electrothermal Average Inductor Model for Analyses of Boost Converters Krzysztof Górecki, Janusz Zarębski, Kalina Detka Gdynia Maritime University Department of Marine Electronics

2. Outline Introduction The average electrothermal model of the diode-transistor switch The average electrothermal model of the inductor Results of calculations and measurements Conclusions

3. Introduction (1) The boost converter is frequently used in switched-mode power supplies. The important parts of this converter are semiconductor devices and an inductor with the ferromagnetic core. In literature the properties of this circuit are analyzed, but the consideration focused only on semiconductor devices ignoring properties of magnetic elements. In previous authors papers it is shown, that the nonlinearity dependence L(i) significantly influences the course of characteristics of the considered converter. In the cited papers, in order to calculate the characteristics of dc- dc converters the classical transient analysis method is used. The analysis is time consuming, and there are often problems to obtain convergence of computations. During dc-dc converters operation the internal components temperature rises due to a self-heating phenomenon.

4. Introduction (2) To take into account this phenomenon in the computer analysis, special electrothermal models are required. The average electrothermal model of the diode – transistor switch for SPICE is proposed, but in this model non-linearity of the inductor characteristics and self-heating in this element are omitted. In this paper the average electrothermal model of the inductor dedicated to the electrothermal analysis of dc-dc converters is proposed. This model takes into account magnetic and thermal phenomena in the inductor and power losses in the core and in the winding. The model is used to calculate non-isothermal characteristics of the boost converter.

5. The average electrothermal model of the diode-transistor switch Control voltage sources Et and Er represent the unipolar transistor, connected to terminals 1, 2 and 5 The diode (between the terminals 3 and 4) is represented by the controlled current source Gd. T he auxiliary circuit is used to determine the mode of dc – dc converter (CCM or DCM) including the considered switch. The aided block model the temperature changes of transistor resistance R ON, series diode resistance R D and voltage U D on the forward biased p-n junction. The internal diode and transistor temperature are calculated in the thermal model.

6. The average electrothermal model of the inductor The average electrothermal model of the inductor refers to an average value of voltage and current in the circuit, instead of the transient values. The values of all the above mentioned quantities can be determined by the electrothermal model of the inductor Terminals A and B are the terminals of the inductor. Terminal L - the voltage corresponding to the inductor inductance, T erminal T U - the temperature of the inductor winding, T erminal T R - the temperature of the inductor core.

7. The average electrothermal model of the inductor (2) In the main circuit: the voltage source V L with zero output voltage monitors the current of the inductor, the resistor R S0 represents the winding resistance of the inductor for the direct current at the reference temperature T 0 the controlled voltage source E RS describes the influence of temperature and losses of the inductor series resistance. the controlled voltage source E RN models the skin effect phenomenon In the auxiliary block the controlled voltage sources are used to calculate the value of: the magnetic force H, flux density B, B sat saturation flux density the auxiliary value C. The thermal model allows calculating an excess of the core T R and winding T U temperature above the ambient temperature T a.

8. Results of calculations and measurements The average electrothermal model of the inductor is used to calculate the characteristics of the boost converter, the inductor L containing 27 turns of enamel copper wire of the diameter of 0.8 mm wound on a toroidal ferrite core RTF25x15x10 made of F867 material. The considered circuit is excited by the DC voltage equal to 12 V, the frequency of the control signal is f = 50 kHz.

9. Results of calculations and measurements (2) curves a - results of analyses obtained using average electrothermal models of the diode-transistor switch and the inductor, curves b – the results obtained using the average electrothermal model of the diode-transistor switch and the linear inductor model without losses, curves c - the results of analyses using isothermal models of considered devices, curves d - the results of the transient analysis obtained using the electrothermal models of the transistor, the diode and the inductor.

10. Results of calculations and measurements (3) It is easy to notice that in the considered range of load resistance the boost converter operates in both CCM and DCM modes. Taking into account non-linearity of L(i) caused the shifts of the boundary between CCM and DCM mode towards bigger values of load resistance R 0. In the range of small values of R 0 it can be seen that using the classical model without losses allows obtaining the constant output voltage U wy and watt-hour efficiency. From both the considered models the increasing dependences U wy (R 0 ) and  (R 0 ) are obtained, Taking into account non-linearity of the inductor model reduces the converter output voltage and watt-hour efficiency. It is worth noticing, that taking into account the inductor non- linearities and losses in the inductor causes the reduction of 50% in the output voltage, and the watt-hour efficiency - as much as 60%.

11. Results of calculations and measurements (4) The considered dependences are decreasing functions. For the high resistance R 0, the core temperature is higher than the winding temperature and for the small resistance R 0 the winding temperature achieves the higher value. The computations time for the presented characteristics using average electrothermal models was about a ten miliseconds, whereas the determination of these characteristics by the transient electrothermal model takes several hours.

12. Conclusions The paper presents the average electrothermal model of the inductor dedicated to determine non-isothermal characteristics of dc-dc converters using SPICE. The proposed model takes into account the dependence of inductance and series resistance on: the inductor current, frequency, temperature inductor construction parameters parameters of the core. The results of computations show that taking into account in the presented model the physical phenomenon clearly affect characteristics of the considered circuit In particular, the shift of the boundary between CCM and DCM mode towards higher values of R 0 resistance is observed.

13. Conclusions (2) The output voltage and watt-hour efficiency of the boost converter in the range of small values of R 0 decrease. The modified model also allows the computation of the core and winding temperature. Simultaneously, the time needed to compute the non- isothermal characteristics of the considered converter is much shorter than using the classical transient analysis and the electrothermal model of the inductor.