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Modelling electronic circuit radiation cooling using analytical thermal model
Microelectronics Journal Microelectronics Journal 31 (2000) 781–785 www.elsevier.com/locate/mejo
Modelling electronic circuit radiation cooling using analytical thermal model M. Janicki*, A. Napieralski Department of Microelectronics and Computer Science, Technical University of Lodz, Al. Politechniki 11, PL-93-590 Lodz, Poland
Abstract This paper presents a thermal model of electronic circuit based on an analytical solution of the heat equation. In particular, the question of modelling circuit radiation cooling is discussed. The proposed model with temperature dependent heat exchange coefficient is applied for the analysis of an integrated circuit. Simulation results are validated with both infrared and p–n junction measurements. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Heat equation; Circuit thermal model
1. Introduction Due to the fact that electronic devices in contemporary circuits are densely packed and dissipate a large amount of power at high operating frequencies, high temperature rises can occur even in apparently low power circuits. The seriousness of the problem is illustrated in the chart below (Fig. 1), which shows the experimental results of a research, which was intended to investigate the causes of contemporary microelectronic circuit failures [1]. Surprisingly, over half of the failures are due to thermal reasons! Taking into account that excessive temperature of semiconductor structure has a strong effect on device lifetime and eventually can lead to circuit thermal failure, thermal management of electronic circuits became a problem of great importance. Thermal models of electronic circuits are necessary already in the early stages of circuit design process. Improper arrangement of power devices in circuit layout can have serious impact on circuit performance. Therefore, during the layout design, all the elements which have major influence on circuit temperature should be placed in such a manner so as to assure, taking into consideration some design constraints, the minimal possible temperature rise. This task can be accomplished developing and employing adequate circuit thermal models. Another area of thermal model application is the real time circuit temperature monitoring. Many modern electronic circuits require continuous on-line monitoring of their temperature either for control or protection purposes. In
this case, the thermal model can be employed to estimate the temperature of the whole circuit based on sensor temperature measurements only at selected points. All the simulations presented in this paper are based on the original thermal model of electronic circuits developed by the authors at The Division of Microelectronics and Computer Science of The Technical University of Lodz. Owing to the simplicity of the proposed model, it is possible to compute temperature distribution in a structure using an analytical Fourier series solution of the heat equation. The next section of this paper describes briefly the problem of heat exchange modelling in solids. This is followed by the description of the thermal model of electronic circuit, which was used for the simulations. Finally, an analytical thermal model of a real integrated circuit is created and validated with measurements. Based on the measurements, the problem of taking into account highly variable with temperature circuit radiation cooling is discussed.
* Corresponding author. Tel.: ⫹48-42-631-2645; fax: ⫹48-42-636-0327. E-mail address: [email protected] (M. Janicki). 0026-2692/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(00)00059-8
Fig. 1. Causes of microcircuit failures.
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M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
the form of the Laplace equation:
2. Heat exchange modelling Heat can be exchanged by means of conduction, convection or radiation. In the case of electronic circuits, generated heat is conducted towards structure edges where it is removed through convection or radiation. The heat conduction process is governed by the Fourier law represented in Eq. (1). According to this law, the heat flux is proportional to the temperature gradient
7 2T 0
4
where h is the heat exchange coefficient (W/K m ), T the surface temperature (K), and T∞ the surrounding fluid temperature (K). The simplicity of the above formula is misleading because the heat exchange coefficient takes account of both radiation and convection and in practice it is extremely difficult to determine the actual value of this coefficient which depends both on surface temperature and its emissivity. Usually, an averaged value of the coefficient is assumed for computations. The complete physical model of the heat exchange in solids is based on the Fourier law of conduction and the energy balance for a unitary volume and can be summarised in the following three-dimensional heat equation:
Given semiconductor material thermal properties, generated heat density, and boundary conditions, temperature distribution in the whole structure can be determined solving the above equation. The methods for solving such partial differential equations can be divided into two general groups: analytical methods and numerical methods. In this publication, the authors focused particularly on analytical methods of solution because they render possible to obtain exact solutions in the form of a single formula for the whole analysis domain. In general, it is difficult to find the exact solution because of the fact that real structures have too complex shapes or boundary conditions. Fortunately, most electronic circuits usually have relatively simple shapes and analytical methods can be applied for their analysis. Then, if a model is not excessively simplified, obtained analytical solutions are more accurate than numerical ones, which inherently are not exact and depend on the choice of discretisation mesh. Namely, numerical methods would yield accurate solutions (subject to machine accuracy), if the structure discretisation mesh were infinitely dense. For the solution of the circuit thermal model, which is presented in Section 3, the authors employed the separation of variables technique using the Fourier series. This method was chosen because power dissipation and structure temperature rise in the final solution are related to each other by a single coefficient, thus the model allows fast computation of circuit temperature at any location. More information on heat transfer can be found in Refs. [2–5]. Outlines of different heat equation solution methods, both numerical and analytical, are given in Ref. [6].
a a 2T gv ⫹ 7l7T l l 2t
3. Electronic circuit thermal model
q ⫺l7T
1
where q is the heat flux (W/m ), l the thermal conductivity (W/K m), and T the temperature (K). The heat removal through convection and radiation at structure surface can be represented by the Newton law, where the heat flux is proportional to the difference of temperature between the media 2
q h T T ⫺ T ∞
2 2
3
where a is the thermal diffusivity (m 2/s), gv the generated heat volumic density (W/m 3), and t the time (s). For homogenous isotropic substances, when thermal conductivity depends only on space co-ordinates, the third component on the left-hand side of Eq. (3) vanishes and the equation becomes linear. Furthermore, if there is no internal heat generation the equation takes the form of heat diffusion equation and only steady states are considered, Eq. (3) takes
This section is devoted to the description of the thermal model of electronic circuit employed by the authors in the simulations presented later in this paper. The adopted approach is not entirely new and is based on some earlier thermal models of electronic circuits [7–9], but the problem was expressly posed in such a way that the form of its solution allows fast estimation of circuit temperature.
Fig. 2. Single layer circuit model.
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
783
Fig. 3. Multilayer thermal model.
As already mentioned, the model developed by the authors is based on the full three-dimensional form of the heat equation and its analytical Fourier series solution. In the model, it is assumed that the thermal conductivity is constant and independent from temperature, which implies that the temperature rise is a linear function of dissipated power. Power dissipation in heat sources placed at the circuit top surface is represented by heat flux penetrating into the structure. Taking into account thin slab geometry of most electronic circuits, the lateral surfaces of the structure are regarded as adiabatic ones and the heat is removed at the bottom, which is modelled by the heat exchange coefficient. The whole model is pictured in Fig. 2. Comparing to the model suggested by Lindsted and Surty [7], in this model the isothermal boundary condition at the bottom of the structure was replaced by the Newton boundary condition, which constitutes a more realistic and general case [10]. Because the above simple version of the thermal model proved to be insufficient in some cases [11], hence it was further developed. The enhancement was accomplished by
the inclusion of additional layers. This required imposing the condition of heat flux equality at imperfect layer contacts which are represented by the interlayer thermal conductance g. Furthermore, the possibility of circuit cooling on the upper surface was also included. The entire improved model, illustrated in Fig. 3, can be summarised in the following set of equations: Governing Laplace equation : 7 2 T 0
5
Boundary conditions: Top surface : ⫺l
2T ⫹ h T ⫺ T∞ q 2z
6
Lateral surfaces :
2T 2T 0; 0 2x 2y
7
Imperfect contacts : li
2Ti 2T 2T li⫹1 i⫹1 ; ⫺li i 2z 2z 2z
g Ti ⫺ Ti⫹1
8
Bottom surface : ⫺l
2T h T ⫺ T∞ 2z
9
where g is the thermal conductance (W/K m 2) i, i ⫹ 1 the layer indexes, and x, y, z the co-ordinates. If the contacts between materials are assumed perfect, Eq. (8) in the model can be rewritten as follows: Perfect contacts : li
Fig. 4. Photograph of the MC 33186 integrated circuit.
2Ti 2T li⫹1 i⫹1 ; Ti Ti⫹1 2z 2z
10
Obviously, the above model cannot give errorless solutions but in most practical cases its accuracy proved to be satisfactory. The errors originate mainly from the adopted model of heat removal. In reality, heat is removed not only at the bottom through a heat sink and at the top via convection and radiation, but also through the chip bonding wires and lateral faces of the structure, which is not included in the thermal model. Moreover, as it will be demonstrated in the next section, the heat exchange coefficient value is temperature dependent, especially for radiation cooling. Additionally, material thermal properties also depend on
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M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
Fig. 5. Schematic layout of the MC 33186 integrated circuit.
temperature, but in a narrow operating temperature range they can be regarded as constant ones. The analytical solution of the model using Fourier series for a single layer can be found in Ref. [10].
4. Experiment The problem of the choice of the heat exchange coefficient value in the thermal model will be demonstrated in this section based on a practical example of the commercial integrated DC motor driver MC 33186. This circuit contains four large TMOS power transistors which are the main heat sources in the circuit. The photograph of the circuit and its schematic layout are presented in Figs. 4 and 5, respectively. In order to collect the data necessary to create a thermal model of the circuit, circuit temperature was measured for different values of power dissipated in transistor T2. The measurements were performed using both the p–n junction technique and the infrared thermography. More on the temperature measurements techniques can be found in Ref. [12]. For the infrared measurements, the semiconductor surface was covered with black matte paint so as to assure uniform surface emissivity. Based on measurement results, the thermal model of the circuit, which is presented in Fig. 6, was created. The model comprises the silicon substrate, in which the transistors are integrated and the copper heat sink soldered to the silicon. The thickness and thermal conductivity of each layer, which were provided by the manufacturer, are also given in Fig. 6. The heat exchange coefficient value for forced water cooling at
Fig. 6. Cross-sectional view of the integrated circuit thermal model.
Fig. 7. Comparison of the linear model simulation with p–n junction measurement.
the bottom surface was set experimentally to fit the experimental data from the p–n junction measurement. As can be seen from Fig. 7 this linear thermal model accurately matches the experimental data. The average error is 0.8 K and its maximal value is 2.6 K. Unfortunately, this simple thermal model turned out to be inaccurate in the case of thermographical measurements with the maximal error value reaching 25.9 K (see Fig. 8). This is due to the fact that these measurements were performed, unlike for the p–n junction measurements, with open IC package. Then, heat can be evacuated through convection and, especially at higher temperatures, radiation, which was not possible with closed package. In order to solve the problem we proposed to include in the model the possibility of circuit cooling on the upper surface. The cooling is modelled by the heat exchange coefficient dependent on temperature, similarly to the model discussed in Ref. [13]. However, taking into account that at lower temperatures the value of this coefficient is much lower than the one at the circuit bottom surface, we decided to remove the constant term, which is usually added to the
Fig. 8. Model comparison with thermographical measurements.
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
coefficient value. Thus, in the improved model the value of heat exchange coefficient was modified with temperature according to the following formula: DT m 11 h h0 DT0 where h0 is the heat exchange coefficient for temperature rise DT0 (W/K m 2), DT the temperature rise (K), and m the exponent. The reference value of the heat exchange coefficient h0 (0.023 W/K cm 2) was set to match the infrared measurement data for the temperature rise equal to 74.7 K. The optimal value of the exponent m was equal to 1.72. This value corresponds rather to the theoretical value for pure radiation cooling m 1:5–1:6 than for combined radiation–convection cooling m 0:6–0:8: Slightly higher value than the theoretical one can result from the neglecting of the constant term. As can be seen in Fig. 8 the simulation results obtained using the improved model with the temperature dependent heat exchange coefficient are considerably better than for the purely linear model. Namely, the average error and its maximal value dropped down to 0.9 and 2.6 K, respectively. It should be emphasised that the improved model gives particularly accurate results at high temperatures where the natural convection at the top surface can be neglected in comparison to the radiation cooling. It is also worth mentioning that the value of the heat exchange coefficient at the top surface, even at temperature reaching 400 K, is equal to only one third of the one corresponding to forced water cooling at the bottom surface. Using Eq. (11), it can be found that theoretically the radiation cooling at the top surface would be equal to the water cooling at the bottom at temperature rise close to 250 K. 5. Conclusions In this paper, it was demonstrated that integrated circuits can be successfully modelled in quite a wide range of operating temperatures using analytical solutions of the heat equation. However then, in most cases, the temperature dependence of some model parameters, such as thermal
785
conductivity or heat exchange coefficient, must be taken into consideration. In the particular case discussed in this paper, in order to account for the radiation cooling, which is strongly temperature dependent, the value of the heat exchange coefficient at the top surface was variable with temperature. Owing to this, it was possible to significantly improve the simulation results. Similar approach can be used for hybrid circuits where radiation plays a considerable role.
Acknowledgements This work was supported by the university internal grant K-25/Dz.St/1/99.
References [1] V.K. Maudgal, Computer-aided thermal analysis, Hybrid Circuit Technology May (1991) 19–21. [2] A.L. Lydersen, Fluid Flow and Heat Transfer, Wiley, New York, 1989. [3] S. Kakac, Y. Yener, Heat Conduction, Taylor & Francis, London, 1993. [4] H.S. Carslaw, J.S. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1947. [5] Y. Bayazitoglu, M.N. Ozisik, Elements of Heat Transfer, McGrawHill, New York, 1988. [6] M.N. Ozisik, Heat Conduction, Wiley, New York, 1993. [7] R.D. Lindsted, R.J. Surty, Steady-state junction temperatures of semiconductor chips, IEEE Transactions on Electron Devices ED-19 (1) (1972). [8] A.G. Kokkas, Thermal analysis of multiple-layer structures, IEEE Transactions on Electron Devices ED-21 (11) (1974) 674–681. [9] Kos, G. De Mey, Thermal Modelling and Optimisation of Power Microstructures, Electrochemical Publications, British Isles, 1997. [10] M. Janicki, M. Zubert, A. Napieralski, Application of inverse heat conduction methods in temperature monitoring of integrated circuits, Sensors and Actuators, A. Physical 71 (1/2) (1998) 51–57. [11] M. Janicki, A. Napieralski, Considerations on accuracy of analytical thermal models of electronic circuits, Proceedings of the Sixth International Conference on Mixed Design of Integrated Circuits and Systems, Krakow, Poland, 17–19 June 1999, pp. 305–308. [12] J.W. Gardner, Microsensors, Wiley, New York, 1994. [13] J.P. Holman, Heat Transfer, McGraw-Hill, New York, 1985.
Modelling electronic circuit radiation cooling using analytical thermal model M. Janicki*, A. Napieralski Department of Microelectronics and Computer Science, Technical University of Lodz, Al. Politechniki 11, PL-93-590 Lodz, Poland
Abstract This paper presents a thermal model of electronic circuit based on an analytical solution of the heat equation. In particular, the question of modelling circuit radiation cooling is discussed. The proposed model with temperature dependent heat exchange coefficient is applied for the analysis of an integrated circuit. Simulation results are validated with both infrared and p–n junction measurements. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Heat equation; Circuit thermal model
1. Introduction Due to the fact that electronic devices in contemporary circuits are densely packed and dissipate a large amount of power at high operating frequencies, high temperature rises can occur even in apparently low power circuits. The seriousness of the problem is illustrated in the chart below (Fig. 1), which shows the experimental results of a research, which was intended to investigate the causes of contemporary microelectronic circuit failures [1]. Surprisingly, over half of the failures are due to thermal reasons! Taking into account that excessive temperature of semiconductor structure has a strong effect on device lifetime and eventually can lead to circuit thermal failure, thermal management of electronic circuits became a problem of great importance. Thermal models of electronic circuits are necessary already in the early stages of circuit design process. Improper arrangement of power devices in circuit layout can have serious impact on circuit performance. Therefore, during the layout design, all the elements which have major influence on circuit temperature should be placed in such a manner so as to assure, taking into consideration some design constraints, the minimal possible temperature rise. This task can be accomplished developing and employing adequate circuit thermal models. Another area of thermal model application is the real time circuit temperature monitoring. Many modern electronic circuits require continuous on-line monitoring of their temperature either for control or protection purposes. In
this case, the thermal model can be employed to estimate the temperature of the whole circuit based on sensor temperature measurements only at selected points. All the simulations presented in this paper are based on the original thermal model of electronic circuits developed by the authors at The Division of Microelectronics and Computer Science of The Technical University of Lodz. Owing to the simplicity of the proposed model, it is possible to compute temperature distribution in a structure using an analytical Fourier series solution of the heat equation. The next section of this paper describes briefly the problem of heat exchange modelling in solids. This is followed by the description of the thermal model of electronic circuit, which was used for the simulations. Finally, an analytical thermal model of a real integrated circuit is created and validated with measurements. Based on the measurements, the problem of taking into account highly variable with temperature circuit radiation cooling is discussed.
* Corresponding author. Tel.: ⫹48-42-631-2645; fax: ⫹48-42-636-0327. E-mail address: [email protected] (M. Janicki). 0026-2692/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(00)00059-8
Fig. 1. Causes of microcircuit failures.
782
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
the form of the Laplace equation:
2. Heat exchange modelling Heat can be exchanged by means of conduction, convection or radiation. In the case of electronic circuits, generated heat is conducted towards structure edges where it is removed through convection or radiation. The heat conduction process is governed by the Fourier law represented in Eq. (1). According to this law, the heat flux is proportional to the temperature gradient
7 2T 0
4
where h is the heat exchange coefficient (W/K m ), T the surface temperature (K), and T∞ the surrounding fluid temperature (K). The simplicity of the above formula is misleading because the heat exchange coefficient takes account of both radiation and convection and in practice it is extremely difficult to determine the actual value of this coefficient which depends both on surface temperature and its emissivity. Usually, an averaged value of the coefficient is assumed for computations. The complete physical model of the heat exchange in solids is based on the Fourier law of conduction and the energy balance for a unitary volume and can be summarised in the following three-dimensional heat equation:
Given semiconductor material thermal properties, generated heat density, and boundary conditions, temperature distribution in the whole structure can be determined solving the above equation. The methods for solving such partial differential equations can be divided into two general groups: analytical methods and numerical methods. In this publication, the authors focused particularly on analytical methods of solution because they render possible to obtain exact solutions in the form of a single formula for the whole analysis domain. In general, it is difficult to find the exact solution because of the fact that real structures have too complex shapes or boundary conditions. Fortunately, most electronic circuits usually have relatively simple shapes and analytical methods can be applied for their analysis. Then, if a model is not excessively simplified, obtained analytical solutions are more accurate than numerical ones, which inherently are not exact and depend on the choice of discretisation mesh. Namely, numerical methods would yield accurate solutions (subject to machine accuracy), if the structure discretisation mesh were infinitely dense. For the solution of the circuit thermal model, which is presented in Section 3, the authors employed the separation of variables technique using the Fourier series. This method was chosen because power dissipation and structure temperature rise in the final solution are related to each other by a single coefficient, thus the model allows fast computation of circuit temperature at any location. More information on heat transfer can be found in Refs. [2–5]. Outlines of different heat equation solution methods, both numerical and analytical, are given in Ref. [6].
a a 2T gv ⫹ 7l7T l l 2t
3. Electronic circuit thermal model
q ⫺l7T
1
where q is the heat flux (W/m ), l the thermal conductivity (W/K m), and T the temperature (K). The heat removal through convection and radiation at structure surface can be represented by the Newton law, where the heat flux is proportional to the difference of temperature between the media 2
q h T T ⫺ T ∞
2 2
3
where a is the thermal diffusivity (m 2/s), gv the generated heat volumic density (W/m 3), and t the time (s). For homogenous isotropic substances, when thermal conductivity depends only on space co-ordinates, the third component on the left-hand side of Eq. (3) vanishes and the equation becomes linear. Furthermore, if there is no internal heat generation the equation takes the form of heat diffusion equation and only steady states are considered, Eq. (3) takes
This section is devoted to the description of the thermal model of electronic circuit employed by the authors in the simulations presented later in this paper. The adopted approach is not entirely new and is based on some earlier thermal models of electronic circuits [7–9], but the problem was expressly posed in such a way that the form of its solution allows fast estimation of circuit temperature.
Fig. 2. Single layer circuit model.
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
783
Fig. 3. Multilayer thermal model.
As already mentioned, the model developed by the authors is based on the full three-dimensional form of the heat equation and its analytical Fourier series solution. In the model, it is assumed that the thermal conductivity is constant and independent from temperature, which implies that the temperature rise is a linear function of dissipated power. Power dissipation in heat sources placed at the circuit top surface is represented by heat flux penetrating into the structure. Taking into account thin slab geometry of most electronic circuits, the lateral surfaces of the structure are regarded as adiabatic ones and the heat is removed at the bottom, which is modelled by the heat exchange coefficient. The whole model is pictured in Fig. 2. Comparing to the model suggested by Lindsted and Surty [7], in this model the isothermal boundary condition at the bottom of the structure was replaced by the Newton boundary condition, which constitutes a more realistic and general case [10]. Because the above simple version of the thermal model proved to be insufficient in some cases [11], hence it was further developed. The enhancement was accomplished by
the inclusion of additional layers. This required imposing the condition of heat flux equality at imperfect layer contacts which are represented by the interlayer thermal conductance g. Furthermore, the possibility of circuit cooling on the upper surface was also included. The entire improved model, illustrated in Fig. 3, can be summarised in the following set of equations: Governing Laplace equation : 7 2 T 0
5
Boundary conditions: Top surface : ⫺l
2T ⫹ h T ⫺ T∞ q 2z
6
Lateral surfaces :
2T 2T 0; 0 2x 2y
7
Imperfect contacts : li
2Ti 2T 2T li⫹1 i⫹1 ; ⫺li i 2z 2z 2z
g Ti ⫺ Ti⫹1
8
Bottom surface : ⫺l
2T h T ⫺ T∞ 2z
9
where g is the thermal conductance (W/K m 2) i, i ⫹ 1 the layer indexes, and x, y, z the co-ordinates. If the contacts between materials are assumed perfect, Eq. (8) in the model can be rewritten as follows: Perfect contacts : li
Fig. 4. Photograph of the MC 33186 integrated circuit.
2Ti 2T li⫹1 i⫹1 ; Ti Ti⫹1 2z 2z
10
Obviously, the above model cannot give errorless solutions but in most practical cases its accuracy proved to be satisfactory. The errors originate mainly from the adopted model of heat removal. In reality, heat is removed not only at the bottom through a heat sink and at the top via convection and radiation, but also through the chip bonding wires and lateral faces of the structure, which is not included in the thermal model. Moreover, as it will be demonstrated in the next section, the heat exchange coefficient value is temperature dependent, especially for radiation cooling. Additionally, material thermal properties also depend on
784
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
Fig. 5. Schematic layout of the MC 33186 integrated circuit.
temperature, but in a narrow operating temperature range they can be regarded as constant ones. The analytical solution of the model using Fourier series for a single layer can be found in Ref. [10].
4. Experiment The problem of the choice of the heat exchange coefficient value in the thermal model will be demonstrated in this section based on a practical example of the commercial integrated DC motor driver MC 33186. This circuit contains four large TMOS power transistors which are the main heat sources in the circuit. The photograph of the circuit and its schematic layout are presented in Figs. 4 and 5, respectively. In order to collect the data necessary to create a thermal model of the circuit, circuit temperature was measured for different values of power dissipated in transistor T2. The measurements were performed using both the p–n junction technique and the infrared thermography. More on the temperature measurements techniques can be found in Ref. [12]. For the infrared measurements, the semiconductor surface was covered with black matte paint so as to assure uniform surface emissivity. Based on measurement results, the thermal model of the circuit, which is presented in Fig. 6, was created. The model comprises the silicon substrate, in which the transistors are integrated and the copper heat sink soldered to the silicon. The thickness and thermal conductivity of each layer, which were provided by the manufacturer, are also given in Fig. 6. The heat exchange coefficient value for forced water cooling at
Fig. 6. Cross-sectional view of the integrated circuit thermal model.
Fig. 7. Comparison of the linear model simulation with p–n junction measurement.
the bottom surface was set experimentally to fit the experimental data from the p–n junction measurement. As can be seen from Fig. 7 this linear thermal model accurately matches the experimental data. The average error is 0.8 K and its maximal value is 2.6 K. Unfortunately, this simple thermal model turned out to be inaccurate in the case of thermographical measurements with the maximal error value reaching 25.9 K (see Fig. 8). This is due to the fact that these measurements were performed, unlike for the p–n junction measurements, with open IC package. Then, heat can be evacuated through convection and, especially at higher temperatures, radiation, which was not possible with closed package. In order to solve the problem we proposed to include in the model the possibility of circuit cooling on the upper surface. The cooling is modelled by the heat exchange coefficient dependent on temperature, similarly to the model discussed in Ref. [13]. However, taking into account that at lower temperatures the value of this coefficient is much lower than the one at the circuit bottom surface, we decided to remove the constant term, which is usually added to the
Fig. 8. Model comparison with thermographical measurements.
M. Janicki, A. Napieralski / Microelectronics Journal 31 (2000) 781–785
coefficient value. Thus, in the improved model the value of heat exchange coefficient was modified with temperature according to the following formula: DT m 11 h h0 DT0 where h0 is the heat exchange coefficient for temperature rise DT0 (W/K m 2), DT the temperature rise (K), and m the exponent. The reference value of the heat exchange coefficient h0 (0.023 W/K cm 2) was set to match the infrared measurement data for the temperature rise equal to 74.7 K. The optimal value of the exponent m was equal to 1.72. This value corresponds rather to the theoretical value for pure radiation cooling m 1:5–1:6 than for combined radiation–convection cooling m 0:6–0:8: Slightly higher value than the theoretical one can result from the neglecting of the constant term. As can be seen in Fig. 8 the simulation results obtained using the improved model with the temperature dependent heat exchange coefficient are considerably better than for the purely linear model. Namely, the average error and its maximal value dropped down to 0.9 and 2.6 K, respectively. It should be emphasised that the improved model gives particularly accurate results at high temperatures where the natural convection at the top surface can be neglected in comparison to the radiation cooling. It is also worth mentioning that the value of the heat exchange coefficient at the top surface, even at temperature reaching 400 K, is equal to only one third of the one corresponding to forced water cooling at the bottom surface. Using Eq. (11), it can be found that theoretically the radiation cooling at the top surface would be equal to the water cooling at the bottom at temperature rise close to 250 K. 5. Conclusions In this paper, it was demonstrated that integrated circuits can be successfully modelled in quite a wide range of operating temperatures using analytical solutions of the heat equation. However then, in most cases, the temperature dependence of some model parameters, such as thermal
785
conductivity or heat exchange coefficient, must be taken into consideration. In the particular case discussed in this paper, in order to account for the radiation cooling, which is strongly temperature dependent, the value of the heat exchange coefficient at the top surface was variable with temperature. Owing to this, it was possible to significantly improve the simulation results. Similar approach can be used for hybrid circuits where radiation plays a considerable role.
Acknowledgements This work was supported by the university internal grant K-25/Dz.St/1/99.
References [1] V.K. Maudgal, Computer-aided thermal analysis, Hybrid Circuit Technology May (1991) 19–21. [2] A.L. Lydersen, Fluid Flow and Heat Transfer, Wiley, New York, 1989. [3] S. Kakac, Y. Yener, Heat Conduction, Taylor & Francis, London, 1993. [4] H.S. Carslaw, J.S. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1947. [5] Y. Bayazitoglu, M.N. Ozisik, Elements of Heat Transfer, McGrawHill, New York, 1988. [6] M.N. Ozisik, Heat Conduction, Wiley, New York, 1993. [7] R.D. Lindsted, R.J. Surty, Steady-state junction temperatures of semiconductor chips, IEEE Transactions on Electron Devices ED-19 (1) (1972). [8] A.G. Kokkas, Thermal analysis of multiple-layer structures, IEEE Transactions on Electron Devices ED-21 (11) (1974) 674–681. [9] Kos, G. De Mey, Thermal Modelling and Optimisation of Power Microstructures, Electrochemical Publications, British Isles, 1997. [10] M. Janicki, M. Zubert, A. Napieralski, Application of inverse heat conduction methods in temperature monitoring of integrated circuits, Sensors and Actuators, A. Physical 71 (1/2) (1998) 51–57. [11] M. Janicki, A. Napieralski, Considerations on accuracy of analytical thermal models of electronic circuits, Proceedings of the Sixth International Conference on Mixed Design of Integrated Circuits and Systems, Krakow, Poland, 17–19 June 1999, pp. 305–308. [12] J.W. Gardner, Microsensors, Wiley, New York, 1994. [13] J.P. Holman, Heat Transfer, McGraw-Hill, New York, 1985.