A Semiconductor Device Thermal Model Taking into Account Non-linearity and Multhipathing of the Cooling System

The paper is devoted to modelling thermal properties of semiconductor devices at the steady state. The dc thermal model of a semiconductor device taking into account the multipath heat flow is proposed. Some results of calculations and measurements of thermal resistance of a power MOSFET operating at different cooling conditions are presented. The obtained results of calculations fit the results of measurements, which proves the correctness of the proposed model.

thermal resistance is clearly visible.
In the literature, such as [15 -19] only the selected aspects of modelling the thermal behaviour of electronic components are considered. Moreover, empirical formulas describing dependencies of the device transient thermal impedance on the dissipated power for one chosen cooling condition of the device are described [3].  In this paper, which is an extended version of the paper [13], thermal properties of the circuit consisting of the power transistor mounted on the aluminium heat-sink with insulating washer and the Peltier module are analysed. Additionally, the thermal properties of this transistor operating without any heat-sink or located on the heat-sink without the Peltier module are considered.
In section 2 the measurement setup used to measure the thermal parameters of semiconductor devices is presented. The form of the thermal model is described in section 3. Section 4 presents the results of investigations of the power MOS transistor operating at different cooling conditions.

Measurement setup
The results of measurements presented in this paper are obtained by means of the electric method of measurements of transient thermal impedance using the cooling curve [11,20]. The voltage on the forward biased body diode is used as a thermally-sensitive parameter. The measurements were realised in the authors' measurement setup, described in the paper [12]. The block diagram of this measurement setup is shown in figure 2. In the presented measurement setup, the examined transistor (DUT) operates in the common gate configuration. The source current is regulated by means of the voltage source V SS and the resistor R S , and its value I S is measured by the ammeter. The value of the gate-drain voltage V GD is regulated by the voltage source V DD . The values of voltages V GS and V GD are measured by means of voltmeters. The measurement of thermal resistance is realised in 3 stages. First, the thermometric characteristic V GD (T) of the body diode at the forward current equal to I M is measured. During the calibration the switch S 2 is off and the switch S 1 -is on. The value of the current I M is equal to 10 mA. At this value of I M the thermometric characteristic is linear [12]. In the second stage, the examined transistor self in it. This stage lasts until the steady state is obtained and the voltage V GS practically does not change. In the third stage the transients of the voltage V GD on the body diode are registered from the moment t = 0, in which the switch S 1 is turnedon and the switch S 2 is turned-off, until the steady state is obtained. The PC registers values of the voltage on the body diode and controls the switches S 1 and S 2 .
On the basis of the measured time course V GD (t), the thermometric characteristic V GD (T) and the value of the power P H at the end of the second stage of the measurement, the course of the transient thermal impedance Z th (t) is calculated using the following formula: where T a denotes the ambient temperature and F -the slope of the thermometric characteristic V GD (T). The thermal resistance R th is equal to the value of Z th (t) at the steady state. Of course, it is possible to measure the value of the thermal resistance using any dc method, but the range of currents and voltages, in which this method could be used is strongly limited [8,14].

Form of the thermal model
In the paper the cooling system of the power MOS transistor containing a heat-sink and a Peltier module is considered. The formulated dc thermal model of the considered system of the general form is shown in figure 3. In this model, the controlled current source G th represents power dissipated in the modelled device, the voltage source V Ta -ambient temperature, whereas non-linear resistors -thermal resistance of the components of the heat flow path.
On the basis of the dependencies specified by the producers describing thermal resistances of individual components of the heat flow path and the measurements made by the authors, the analytic expressions describing the dependence of the thermal resistance of individual components of the heat flow path are formulated. The thermal resistance R thj-c is constant and its value is equal to the value given by the producer in the catalogue card. The values of the thermal resistance between the case and the surrounding R thc-s is estimated from the measurements of R thj-a of the transistor operating without any heat-sink as the difference between R thj-a and R thj-c .
In turn, the thermal resistance of the isolating washer R thc-a is usually very small and with the correctly performed connection does not exceed typically 1 K/W. Therefore, in the thermal model one can arbitrarily accept the top value of this parameter R thc-a = 1 K/W. The thermal resistance of the isolating washer can be given on the basis of the values of the thermal conductance and geometrical dimensions of the element case given by the producer or based on the measurements of the thermal resistance R thj-a of the transistor mounted on the heat-sink for two cases: R th1 -without the isolating washer and R th2 -with such a washer. The difference R th2 -R th1 is equal to the value of the thermal resistance R thc-a .
As it was shown e.g. in [21,22] the thermal resistance of the power transistor can change in an essential manner e.g. in the function of the pressure exerted on the washer isolating the heat-sink and the device case. The value R thj-a of the examined device in the considered range of changes of pressure p changes even twice, but for the value of the pressure p higher than 1.2 N/mm 2 , which corresponds to tight screwing of the screw, the thermal resistance is practically constant. Then, for the correctly performed connection of the device to the heat-sink the value of R thc-a is practically constant. In turn, the thermal resistance between the heat-sink and the surrounding R thr-s can be obtained from the catalogue dependences of the thermal resistance of the heat-sink on its length. In the case, when such dependence is not accessible, the value R thr-s can be obtained on the basis of the measurements of the thermal resistance R thj-a of the transistor installed on the heat-sink. For this situation, typically the value of R thc-s is omissible high. In such a case the value R thr-s can be calculated from the dependence: It should be taken into account that as it results e.g. from [12], the value of R thr-s depends on the spatial orientation of the heat-sink and the values of this parameter at different orientations of the heatsink can differ from one another even by about 20%.
The value of the thermal resistance between the Peltier module and the heat-sink R thp-r can be obtained from the measurements of R thj-a of the transistor situated on the heat-sink for two cases: with the Peltier module R th3 and without the Peltier module R th4 . Then The value R thp-r is a decreasing function of both the power p th dissipated in the examined transistor and the power p p supplying the module. Of course, on the basis of the measurements of the mentioned dependences it is possible to describe thermal properties of the selected Peltier module.

Results
In order to verify the correctness of the proposed thermal model, the measurements and calculations of the thermal resistance of the power MOS transistor IRF530 operating at different cooling conditions are performed. The considered thermal model has the form shown in figure 3. On the basis of the measurement performed by the authors, the following values of model parameters were estimated: the thermal resistance between the inside and the case in compliance with the catalogue card [23] is R thj-c = 1.9 K/W. The thermal resistance of the connection between the case and the heat-sink filled by silicone is equal to 0.5 K/W.
Several cooling system configurations are explored. In the simplest case, when the transistor operates without the heat-sink, its thermal resistance is a function of the dissipated power [3, 12,13]. For the vertically situated transistor in the case TO-220, the dependence R thc-a (p th ) can be described with the following empirical formula For example, for the transistor situated on the heat-sink without the additional isolating washer the thermal resistance between the heat-sink and the surrounding R thr-s is non-linear, i.e. its value depends on the power p th dissipated in the transistor according to the formula For the considered cooling system R 0 = 1.9 K/W, R 1 = 1.5 K/W, p 0 = 12 W, a = 8 W.
The operation of the transistor IRF530 separated from the same heat-sink through the Peltier module is also considered. The thermal resistance of the connection between the case of the device and the module is 0.5 K/W. In turn, the thermal resistance of the Peltier module depends both on the temperature difference between its both sides, and hence on the power p th dissipated in the examined device, and on the power p p supplying the Peltier module. The thermal resistance R thp-r is described by the formula For the considered transistor, the values of the parameters in the equation (6) are as follows: R 2 = 4.7 K/W, R 3 = 1.8 K/W, R 4 = 1 K/W, p 1 = 5 W, b = 4 W, c = 3.2 W.
In figures 4 -7 the calculated and measured dependences of the thermal resistance between the interior of the investigated transistor and the surrounding on the dissipated power (figures 4-6) or on the power supplied to the Peltier module (figure 7) at different cooling conditions of the considered transistor, are presented. In these figures points mark the results of measurements, while lines -the results of calculations obtained on the basis of the model proposed in this paper. Figure 4 refers to the transistor operating without any heat-sink. In this case a dominant component of the thermal resistance of the transistor is the thermal resistance between the case and the surrounding R thc-a . The increasing character of the dependence R thc-a (p th ), which testifies the dominant role of the thermal conductivity in dissipation of the heat, determines the shape of the obtained curve. As it can be noticed, the range of changes of the value of the thermal resistance of the transistor is in the considered case not high and does not exceed 10%. In turn, figure 5 refers the transistor situated directly on the heat-sink. In comparison to the values obtained for the transistor without any heat-sink the almost tenfold fall in value of the thermal resistance of the investigated transistor is obtained. In the considered case, the thermal resistance between the case and the surrounding R thc-a is omissible large in relation to the thermal resistance between the heat-sink and the surrounding R thr-a . The decreasing character of the dependence R th (p th ) shows that convection is a dominant mechanism of heat abstraction from the transistor. In both the considered cases the visible fall in the value of thermal resistance at an increase of the power dissipated in the transistor and the power supplying the Peltier module is obtained. It is worth noticing that the use of the not supplied Peltier module caused an increase over twice higher in the value of thermal resistance. It is the result of the introduction of the additional component of thermal resistance between the case of the device and the heat-sink. The forced cooling of the transistor by the use of the Peltier module allows decreasing the thermal resistance of the transistor even by about 20%, but it remain constant on the considerably higher level than in the case, when the Peltier module is not used. As it can be noticed, a satisfied agreement between the measured and calculated dependences R th (p th ) and R th (p p ) is obtained in the range of high value of dissipated power. It testifies to the correctness of the proposed thermal model and the correctness of the estimation of the parameters values of this model.

Conclusions
In the paper the thermal model of power semiconductor devices taking into account the multipath and nonlinearity of the phenomena describing the heat transport is proposed. The detailed considerations are presented on the example of the power MOS transistor IRF530. The results of calculations and measurements shown in the paper prove the correctness of the proposed model. The parameters of this model are estimated on the basis of the catalogue data or from the simple measurement, which can be performed by the user. The obtained results of the verification of the model both at free and forced cooling testify to the generality of the proposed model. The proposed model describes thermal properties of investigated device at the steady state only. Currently the authors are conducting works the aim of which is to take into account in the presented model: thermal inertia, sizes and the shape of the heat-sink or printed circuit boards and the influence of the case of the device on efficiency of heat dissipation in the examined transistor.