IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006
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Electrothermal Simulation of Multichip-Modules With Novel Transient Thermal Model and Time-Dependent Boundary Conditions York C. Gerstenmaier, Member, IEEE, Alberto Castellazzi, Member, IEEE, and Gerhard K. M. Wachutka
Abstract—The ability of monitoring the chip temperatures of power semiconductor modules at all times under various realistic working conditions is the basis for investigating the limits of the maximum permissible load. A novel transient thermal model for the fast calculation of temperature fields and hot spot temperature evolution presented recently is extended to include time-dependent boundary conditions for variations of ambient temperature and surface heat flows. For this a Green’s function representation of the temperature field is used. Also, general initial temperature conditions are included. The method is exemplified by application to a dc/ac converter module for automotive hybrid drives. The thermal model, which can be represented by a thermal equivalent circuit, then is combined with an electrical PSpice-metal-oxide semiconductor field-effect transistor (MOSFET) model to allow for the fully self-consistent electrothermal circuit simulation of 42-V/14-V dc/dc-converter modules. 670 converter periods with altogether 8000 MOSFET switching cycles in the six-chip module can be simulated within 1-h computing time on a Pentium PC. Various simulation results are presented, which demonstrate the feasibility of the simulation method and allow for the optimization of converter losses. Short circuit modes of converter operation are investigated with a high temperature increase also revealing the thermal interaction between different chips. Index Terms—DC- and ac-converter for automotive, electrothermal simulation, multichip modules, PSpice metal-oxide semiconductor field-effect transistor (MOSFET) model, thermal equivalent circuit, time-dependent boundary conditions.
I. INTRODUCTION
A
METHOD for a fast calculation of the complete three-dimensional (3-D) temperature fields in electronic systems has been presented [1], [2]. The method was applied to power semiconductor multichip modules (MCMs) with rapidly varying chip heat source strength. In such cases, computation time with finite element (FE) or other numerical methods [3]–[6] becomes too excessive for problems with large time scales, because too many temperature cycles have to be calculated. With the new method and for the purely thermal simulation the dissipated powers of the chips, deduced from Manuscript received February 18, 2005; revised June 24, 2005. This work was presented in part at the 10th International Workshop on Thermal Investigation of ICs and Systems (THERMINIC), Sophia Antipolis, France, September 2004 and at the 10th MIXDES Conference, Lodz, Poland, June 2003. Recommended by Associate Editor J. A. Ferreira. Y. C. Gerstenmaier is with the Corporate Technology Department, Siemens AG, Munich D-80290, Germany (e-mail:
[email protected]). A. Castellazzi and G. K. M. Wachutka are with the Institute for Physics of Electrotechnology, Munich University of Technology, Munich D-80290, Germany. Digital Object Identifier 10.1109/TPEL.2005.861116
measurements, were external input to the simulation. It was thus possible to calculate the whole 3-D-temperature field for, e.g., a power metal-oxide semiconductor field-effect transistor (MOSFET) dc/dc or dc/ac converter module for any chosen time point, even after millions of converter cycles, in less than 100 s. To achieve this performance the thermal problem was reduced to the linear heat conduction equation. In a fully selfconsistent simulation the electrical behavior of the chips with their dissipated powers have to be calculated by an additional electrical or semiconductor model providing the input for the thermal model which has to be solved simultaneously. The dissipated powers in turn depend on the local chip temperatures in a nonlinear way, so that a nonlinear thermal problem arises. To solve this problem efficiently a reduced thermal equivalent circuit has been proposed [1] for the description of the thermal evolution of arbitrarily chosen “hot spots” of interest, usually the chip centres. The circuit corresponds to an exact translation of the analytical thermal model for the temperature field. Numerous compact static [7]–[11] and transient [12]–[16] thermal models have been established for a fast calculation of temperatures. The notion of “compact” thermal model usually implies boundary condition independence (BCI) [7], [8], i.e., the model is valid for all (or nearly all) reasonable temperatures, heat flows and also heat transfer coefficients applied to the thermal contact areas. An important advantage of the new model presented here is its ease of parameter determination by simple linear least square fit to measured or simulated heating curves (thermal impedances). In this work, the thermal model of [1] will be extended to time-dependent but linear, i.e., temperature-independent boundary conditions, providing a solution to the heat conduction problem in arbitrary environment. This is derived with the help of a Green’s function representation of the temperature field. In addition to the thermal model with homogeneous boundary conditions (bound.c.) presented previously [1], the model allows for time-dependent surface heat sources and ambient temperatures at an arbitrary number of thermal contact areas with nonuniform heat transfer coefficients. From this a reduced thermal model is obtained by reducing the number of field points considered to a few locations of interest. Also reduced thermal equivalent circuits for electrothermal simulation can be established in a rigorous and systematic way. General initial temperature conditions can be included in the model and thermal circuit. Temperature-dependent bound.c. were treated in [17]. Because of the essential nonlinearities, in this case another algorithm with considerably larger calculation times was introduced making use of a boundary grid. Sometimes nonlinearities of
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006
Fig. 1. FEM-simulation of 42-V/14-V dc/dc-converter module with six power MOSFETs; 39 481 grid points.
being the local heat transfer coefficient dewith pending on local boundary temperature and local ambient temperature . In this paper, will be restricted to the linear, i.e., temperature-independent case, so that will only depend on position. denotes the outward normal derivative on the surface . is a surface heat flux source, which is balanced by the heat flow through heat convection in the outward direction and heat conduction into the volume. For 0 and 0, (2) includes homogeneous Neumann or adiabatic conditions. is introduced as solution The Green’s function of the special inhomogeneous problem
can be expressed with the help of
as [17], [19]
Fig. 2. Steady-state, linear FEM-simulation (44104 grid points) of dc/ac-converter (ISG). 24 MOSFETs on six substrates (switches) heated with uniform power. Temperatures in K.
the internal material parameters become important. Especially, thermal conductivity in semiconductors may depend strongly on temperature. In power electronic set-ups as, e.g., displayed in Figs. 1 and 2, the silicon chips constitute only a thin layer of approximately 150 m of the total module-thickness of 7 mm, so that this nonlinearity for the seven-layer structure is not as important as in microelectronics for silicon and GaAs devices. In [18], the effect of temperature-dependent thermal conductivities and specific heat has been investigated for multilayer structures. According to this the maximum error, when using linear models with averaged material parameters amounts to less than 3% for a temperature rise below 80 C.
(3)
for the temperature field . The symbol is used to denotes denote the 3-D-position vector (x,y,z). the nonlinear, i.e., temperature-dependent, heat source (heat may describe, generation density) in the system volume . e.g., the dissipated power evolution of module chips or chip cells at position which generally not only depends on but also on the local temperature . denote the mass density, the specific heat and the thermal conductivity in the structure, respectively, depending on the position . denotes the linear differential operator of the heat conduction equation. The following nonlinear, i.e., temperature-dependent Robintype (mixed) bound.c. on the system surface was used in [17]
As was derived in [17], temperature-dependent (chip) heat sources can be included in (3). Since no restrictions apply concerning the boundary conditions, also nonlinear bound.c. for and are allowed. However, in this work only linear, i.e., temperature-independent bound.c., are considered. This is a good approximation for modules of the type shown in Figs. 1 and 2 when, e.g., strong direct water cooling is applied, so that the nonlinearities of radiative cooling and convection to air can be neglected. Usually the bound.c. of the Green’s function are chosen such 0 for those , where is unknown that for those and known (Dirichlet bound.c.), and , where is unknown and known (Neumann bound.c.). On that part of , where mixed bound.c. (2) apply, mixed bound.c. for can be used. Pure Dirichlet conditions for cannot be included in the mixed conditions (2), because setting leads to , but the product is indeterminate. Therefore, if pure Dirichlet conditions have to be applied on some parts of the surface , the boundary condition is: with for for that part of . It is a difficult problem to construct Green’s function with appropriate bound.c. for complex systems. In our case mixed or adiabatic bound.c. similar to (2) are used for but differing from (2) the bound.c. are homogeneous
(2)
(4)
II. THERMAL MODEL DEFINITION The starting point for the thermal model is the heat conduction equation
(1)
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with , . By virtue of (2), the expression for the integrand in the surface integral of (3) is
(5) Also, other choices for and the integrand of the surface integral are possible, which are discussed in [17]. Using (5), an explicit solution for the field is obtained, provided that is independent of the temperature (linear problem). Otherwise, a nonlinear integral equation for results from (3) which can be directly solved, e.g., by the algorithm suggested in [17]. In this work, for a fast calculation of temperatures over many cycles an indirect solution is pursued. With the help of (3) and using a formal eigenfunction representation of the following model for can be established, as will be shown in Appendix A:
(6) or
with “
” denoting the trace of a matrix . and are low order matrices. The matrix being the transpose of ( , , ) is defined by the following sub-matrices:
with dissipating power
for chip ,
for applied average temperatures areas 1
1
.
at thermal contact
where is the heat flux average at thermal contact . Introducing the generalized sources model (6) can be written as
(7)
Fig. 3. Thermal impedance for a location at the bottom of the module (Fig. 2). Unit step of ambient temperature applied at the bottom side with = 6000 W=(Km ). Congruent lines of FE-simulation and linear model (6).
. In the most general with case of electrothermal simulation the sources may also depend on local temperatures, e.g., chip-temperature for . Equation (7) then becomes an implicit nonlinear equation for . The effective time constants are a small set of 8 to 20 approximately logarithmically distributed numbers, which approximate the infinite set of exact time constants is typically between according to (12). The range of the 10 s and 1000 s for the size of multichip modules to be investigated, as has been discussed in [1], [2]. Apart from this, there is a large amount of . Different sets of will arbitrariness in defining the do equally well in providing a very accurate approximation (6) for (3). Heat sources of much smaller size than the ones in Figs. 1 and 2 may necessitate smaller time constants, because in order to resolve small details spatially, eigenfunctions with high eigenvalues 1 (inverse time constants) are necessary corresponding to Fourier-sine-functions with high wave vector [6]. Large heat sources make it possible to reduce the set of effective time constants to a small set by neglecting small time constants. In [17] small and large heating regions were treated simultaneously with inclusion of all time constants by use of an exact Green’s function. If the summation-limits , , and in (6) and (7) are sufficiently small, is represented by the product of two low order matrices and , the first one depending only on , the other one only on . This allows for a fast calculation of for any time with general linear boundary condition, if the convolution integrals in depend on prescribed input functions. In this case has only to be calculated once and not for every location . This makes possible the calculation of the complete temperature field, even after millions of power cycles, once the model matrix has been established [1]. , , and matrices are obtained by The linear least square fit to finite element method (FEM) simulated heating curves (thermal impedances or unit step responses) for the different single source heating conditions. Details on the algorithm are presented in Appendix B. A very accurate approximation is obtained by using 20 constants , as is exemand for surface plified by Figs. 3–5. The matrices sources are established in the same way as the for volume
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006
Fig. 4. Thermal impedance for a location at module top side (Fig. 2). Unit step of ambient temperature applied at the bottom.
Fig. 5. Nonlogarithmical detail of Fig. 4; Zth from FEM linearly interpolated, model fit oscillatory near zero.
heat sources by adjusting the matrix values to thermal impedances for unit steps of temperature or heat flow applied to the thermal contacts of the device. The derivation of the model (6) from the Green’s function representation (3) is essential to insure that the model (6) can also be used for general time-dependent and temperature-independent bound.c., after having established the model matrices , , and by fit for the special constant boundary conditions. The convolution integrals for , , and are evaluated rapidly by analytical formulas for piecewise linear input , , and [1], [20]. functions III. THERMAL SIMULATION RESULTS The model presented in the last section has been exemplified for less general “zero” or homogeneous bound.c. in [1] by thermal investigation of a 42-V/14-V dc/dc-converter developed for automotive applications [21]. Fig. 1 shows the power electronics module of the converter with six MOSFETs in steadystate operation. In realistic operation, the MOSFETs generate pulses on the scale of 200 ns within periods of 6 s to 0.1 ms. The complete 3-D thermal state of the module after, e.g., 150 s (25 10 converter periods) or at any other time can be computed with the new method in less than 100 s, once the model matrices have been established by fit to FEM-simulation. The resulting graphics are displayed in [1] and are similar to Fig. 1. The extraction of the model matrices from FEM-simulation can be done in less than 15 min for all 39 481 grid points of the FEM-model on a PC with the algorithm of Appendix B. For the FEM-simulation of the module in Fig. 1 only linear bound.c. (2) was used with side walls and topsides being adiabatic. At the bottom side (2) with prescribed constant and was used, being set to zero. For the set-up in Figs. 1 and 2, may be interpreted as the variable temperature of the
Fig. 6. Linear model (6) result for dc/ac-module (Fig. 2) for one chip centre. Chips heated according to full load PWM-operation. Additional periodic change of 10 C of ambient T applied at the bottom side.
water coolant at the module bottom side at position and time . An example for time-dependent ambient temperature in the linear case (constant ) is provided in Fig. 6. The heating curves (thermal impedances ) are created by six FE-computations for the different chip heating cases. One additional FE-simulation for a unit step in ambient (coolant) temperature applied at the module bottom side with prescribed heat transfer coefficient allows to include general time variations of the ambient temperature. The module temperature at the bottom side may vary appreciably with position, but the ambient temperature will be chosen to vary only with time. The 42-V dc/ac-converter or integrated starter generator (ISG) module of Fig. 2 operates in automotive applications together with the dc/dc-converter [1], [22]. Each of the six direct copper bonding (DCB)-substrates with four MOSFETs constitutes one switch of the three-phase dc/ac-full-bridge-converter and is also used as freewheeling diode by virtue of the MOSFET intrinsic -body, -bulk diode (see Fig. 10). On the bottom-side of the module base plate direct water cooling is applied (not shown in Fig. 2). The coolant of the combustion engine may be used for this purpose. Fig. 2 is a modeling example for a high temperature electronics application with extremely high cooling liquid temperature of 400 K. Because of the strong direct water cooling the heat transfer coefficient between module bottom side and coolant is in good approximation temperature-independent, and the thermal problem is thus linear. curve of a location at the As an example the heating— module bottom side adjacent to the coolant is shown in Fig. 3 for a unit step in ambient (coolant) temperature applied at the bottom side. The picture contains both, the original FEM result and the model fit. The lines are indistinguishable in the plot resolution with an error below 1%. Fig. 4 shows the heating curve for a different location, the second MOSFET centre on the fourth substrate on the module top side, separated by 6.4 mm from the coolant whose temperature rises again by 1 K. Due to the heat spreading effect there is some time delay in the chip temperature rise now, which below 0.1 s is small enough to be neglected. The relative error of the fit in this time domain can be large, but is of minor importance, because of the small absolute value. Fig. 4 shows this time domain on a nonlogarithmic scale. Values of close to 0 are approximated by small amplitude oscillations of the fit function. The -curves shown are qualitatively very similar to the chip heating cases with “zero” bound.c. as have been displayed for this and other modules in [1] and [23].
GERSTENMAIER et al.: ELECTROTHERMAL SIMULATION OF MULTICHIP-MODULES
The water coolant temperature usually does not stay constant. For demonstration, a periodic change in temperature is assumed with a period of 5 s and a temperature sweep of 10 C according to 5 C 1 2 5 s turned on at 0. At the module top side the average temperature in steady-state operation of a chip with chip heat sources turned off is half the tem[2]. perature sweep of In Fig. 6, in addition to the ambient temperature sweep also the chip heat sources are turned on. A constant average load current of the three phases of 450 A is assumed for simplicity. In this simplified case, the MOSFETs are used alternately as switches and diodes. A full pulse width modulation (PWM) calculation could also be done at the expense of preparing more complicated input for the time-dependent dissipated powers. IV. REDUCED MODELS AND CIRCUITS In [1], it has been pointed out that by selecting a freely can be reduced chosen set of grid points , the model without loss of accuracy to a reduced model for a description of the temperature evolution at the selected positions only. The model of [1] was only valid for constant homogeneous bound.c. (zero bound.c.). By virtue of (6) the model can be extended to include the effects of varying surface or ambient temperature and varying heat flows at the thermal contact areas. The reduced model thus obtained is valid for all bound.c. applied at the thermal contact areas and therefore could be called a “boundary condition independent model” (BCI-model). However, the notion of BCI and “compact” models as used in the literature (e.g., [7] and [8]) usually implies the use of arbitrary heat transfer coefficients as external bound.c. parameters independently of the model parameters. In our case, the model parameters depend on the , and new matrices have to be determined, when changes. On the other hand the compact models of [7], [8], [10], and [11] deal with small packages with small thermal contact areas in comparison to the multichip modules of Figs. 1 and 2. Those modules always have a large temperature variation along the bottom side of the module base plate depending on the different chip heating cases, also when a constant ambient temperature is applied. Thus, the module can hardly be approximated thermally by a compact model which is independent of . The model (6) uses the special initial condition of homogeneous zero temperature at 0. For inhomogeneous starting temperature the model can easily be extended. A homogeneous starting temperature 0 at can be introduced by simple addition of on the right hand side of (6). In that case, also in (3), can be added and the first volume integral can be discarded, when simultaneously replacing over by in the surface integral of (3). For inhomogeneous starting temperature , the first integral in (3) has to be taken into account in order to create the model. With the help of (13), Appendix A, this term for 0 leads to an additional term on the right hand side of (6) (8) with
Fig. 7. Thermal equivalent circuit with M pairs R Modified heat sources P (t ; T ) according to (9).
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= 1, C
= ef f .
Thus, the separation of and is maintained. Since the and defined in Appendix A are not known directly, the have to be obtained for given by fits to FEM simulated cool down curves for zero loads (turned off heat sources) and zero bound.c., similar as the other model matrices , , and are obtained for heating up processes. In [1], a reduced thermal equivalent circuit has been presented denoting the local temperin the form shown in Fig. 7 with atures of interest. denotes a constant external reference temperature usually chosen to be zero. Since the model (6) can also be derived for electrothermal problems, where the volume heat sources depend on the local heat source temperatures which have to be calculated self-consistently, the circuit in Fig. 7 is also well suited for this case. The modified heat sources in Fig. 7, corresponding to current sources in an electric circuit, also depend on the temperatures
Every pair , , with 1 and 1, 1 , in the Foster type circuit is independent of and gives rise to one basis function 1 in the expansion for the unit step responses [(17), Appendix B]. circuits of this kind have to be solved simultaneously for the heat source temperatures , but generally any number of temperatures at additional locations can be included in the reduced model. So far the reduced model and circuit of Fig. 7 is only valid for “zero” or homogeneous bound.c. In order to extend the model to the general bound.c. valid for (6), the same circuit as in Fig. 7 can be used with the modified heat sources (9) now including with in addition the , matrices and the generalized sources introduced in (7). The proof is obvious by rewriting (7) with the definition (9)
(10) 0. Equation (10) can be exactly represented with again by the circuit of Fig. 7 with modified heat sources (9). Also, the inhomogeneous initial temperature field according to (8) can be included in this circuit description. The circuit simulation in this case has to start at 0 with modified heat sources 1 which are zero for 0.
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Fig. 8. Assembly of thermal system network from subsystems 1 and 2. denotes the heat flow at thermal interface node c.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006
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for 0 is described by (9). This gives rise to an additional term for . We thus have obtained a rigorous and general method to construct transient thermal equivalent circuits for the reduced thermal models introduced in the beginning of this section. For general sources the circuit is of small size with pairs 1, , which is times less than for Foster circuits obtained by direct series connection for the different sources [1]. Other more intuitive methods for arriving at thermal equivalent circuits for packages or other set-ups have been presented, e.g., in [7], [8], [10], and [11] for steady-state models and in [12]–[15] for transient models. The advantage of the network proposed here is its invariance concerning the number of thermal ports (thermal contact areas), which only influences the modified heat sources (9). Other advantages of the circuit are the small size (for small ), simplicity, and, perhaps most important, simple parameter determination for by linear least square fits to FEM results or measurements. Together with an electric circuit, which provides the dissipated powers of the devices by multiplication of load currents and voltages, a full electrothermal simulation of an electronic set-up can be performed. A potential disadvantage of the circuit may be that it is not straight-forward to assemble complex systems from subsystems. Usually this is done by connecting the thermal contacts of the constituent sub-circuits to obtain the equivalent circuit of the full system. This is schematically depicted in Fig. 8. The circuit of Fig. 7 has zero heat current at both ends and , so that the connection of two different sub-circuits at , representing an interface node, will not lead to a heat flow along this interface node. Nevertheless, it is also possible with the new model to assemble systems from subsystems. Let denote the model matrix for a subsystem 1 and denotes the model matrix for a subsystem 2. Both subsystems are connected at thermal contact nodes with heat current into system 1 and into flows system 2. are unknowns. The equations for the complete system are given by circuits of Fig. 7 or (10) for the nodes of system 1, , with replacing in the modified heat sources (9) to give . The nodes of system 2, , are obtained by replacing with in (9) to give . The . sums (9) then include the interface heat currents To obtain an additional equation for the unknown the (10) for and at the interfaces can be subtracted to give by virtue of a total of equations 0 with . This equation can be represented by the circuit of Fig. 9 with grounded node (zero) on the right hand side and an additional constraint of zero temperature on the left hand side. This constraint cannot be realized by grounding the
Fig. 9. Additional thermal equivalent circuit for heat flows subsystems contained in modified sources P (t ; T ).
J
between
Fig. 10. DC/DC converter circuit with DMOS-FET structure including p-body, n-bulk freewheeling diode.
left hand side node, since this would give rise to an unwanted circulating current, which would not correspond to the model equations. The fulfilment of this constraint is achieved by contained in the . The adjusting the interface currents equation resulting from Fig. 9 can in principle be solved for the by deconvolution and has to be solved simultaneously with the other (10) or circuits of Fig. 7 for and . In this work, no use is made of assembling systems from subsystems, since the model matrices for the complete system can be directly created by fit to FEM results. V. ELECTROTHERMAL SIMULATION The practicability of the new thermal equivalent circuit of Fig. 7 within a full electrothermal simulation will now be demonstrated for the dc/dc converter module introduced in Section III. Fig. 1 shows the power electronics module of the converter and Fig. 10 the circuit, which allows for a bidirectional energy flow between the two voltage levels [1], [21]. If the “high side” switch HS is turned on and off and the low side switch LS permanently turned off, the converter works as a buck converter (step-down converter). The energy flow (current flow through inductor) is from the 42-V side to the 14-V side. The lower switch is used as freewheeling diode, which is reverse biased each time the high side switch is turned on. Fig. 10 also shows one cell of the power DMOS-FETs, where the -body and -bulk region constitutes the freewheeling diode in parallel to the MOSFET. When energy is transferred from the 14-V side to the 42-V side, the low side switch is turned on and off, while the high side is permanently turned off and used as freewheeling diode. The converter then works as boost converter (step-up converter). Three converters of the type of Fig. 10 are put in parallel in the module working with a phase delay of 1/3 period and 2/3 periods compared to the first converter. Thus, the voltage ripple on the output side is reduced. Also, the current ripple is reduced which is important in order to diminish losses due to
GERSTENMAIER et al.: ELECTROTHERMAL SIMULATION OF MULTICHIP-MODULES
parasitic resistances in the capacitor on the dc-output side. Ad-filters on the input and output side (not shown in ditional Fig. 10) reduce the voltage ripple further. The application example considered here is for the energy flow from 42 V to 14 V. Originally the dc/dc-converter period amounts to 6 s corresponding to 170-kHz operating frequency in order to keep the voltage ripple low. Thus high losses are produced which originate nearly solely from turn-on and turn-off losses. These losses can be reduced considerably by lowering the converter frequency at the expense of increased , denote voltage ripples on the output side. Let the MOSFET turn-off/on losses at current . denotes the converter period 1 reduced by the very short turn-off/on , 1 . With the duty ratio times defined as MOSFET on-state time divided by , the losses of the high side switch in Fig. 10 within one period can be written , as where is the MOSFET on-state voltage at current and the high side voltage (42 V) during the blocking state with current . Since is very small, the last term can be neglected. The losses of the low side switch (diode) are 1 with the reverse recovery losses and the diode on-state losses . Thus, the losses of both switches divided by the period 1 gives the dissipated power , or
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Fig. 11. Measurement and PSpice simulation of 75-V/80-A OptiMOS MOSFET in diode reverse recovery mode.
Fig. 12. PSpice electrothermal simulation of dc/dc-converter module with 6-s period. Losses of second chip working as high side switch. Converter turn on at t 0.
=
In all practical cases 1 and only the first term involving the switching losses rises linearly with . Hysteresis—losses in the dc/dc-inductor and losses in the capacitors are not taken into account by this formula for . For many purposes it may suffice to use approximated expressions for , , etc. by linearized functions in and applied dc-link voltage. Resulting chip losses can then be averaged over the switching cycles for steady-state operation of the converter and the average power is input for a static thermal resistor or a Foster-type thermal – network attached to each heat source [24]. The method presented here includes precisely the thermal interaction of the different chips in the module by use of the simple Foster-type network of Fig. 7 with the generalized heat sources 1, (9) and the same thermal – constants for all chips. Generally the switching and on-state losses of the MOSFETs (or other devices) not only depend on the converter current , but also on their local temperatures, which for a detailed investigation necessitates full electrothermal simulation. The electric MOSFET sub-circuit model used in the electrothermal simulation is for an Infineon 75-V, 80-A OptiMOS (SPP80N08s2-07) MOSFET. MOSFET models for the circuit simulators PSpice and Saber are available from the company’s website [25] and produce the detailed switching wave forms for turn on and turn off (see Fig. 11). The Infineon level 3 model provides high accuracy and includes temperature-dependence of the electrical characteristics due to selfheating, also for nonconstant temperature. In order not to slow down the computation over many converter cycles too much, a modified model was used by us. In particular, constant values were used for the internal capacitance components, short-channel effects (i.e., drain-induced barrier lowering) were neglected and the values
of parasitic inductances and resistances were reduced to a negligible level. Nevertheless the simulation time for the electric MOSFET subcircuit part is still high in comparison to the negligible computation time for the thermal network part. In the following, only the results for the first two chips in the module are considered, since the results for the other chips are very similar. Fig. 12 shows an electrothermal simulation result of the complete dc/dc-converter module with a modified Infineon, level 3 model for all six MOSFETs and with a thermal 11 effective equivalent circuit corresponding to Fig. 7 with time constants . The simulation was done using PSpice [26]. The reduction of the number of from 20 in the purely thermal simulation [1] for the converter in Fig. 1 to 11 in this work is done for reasons of convenience for input to the PSpice simulator. The accuracy of the model fit to the FEM simulated thermal impedances is reduced by lowering . However, the result with 11 is still nearly the same within the plot resolution. Larger errors can arise by separated locations of the monitored temperature and of the heat source. Fig. 12 displays the losses with peaks at the switching instances of the second chip working as high side switch with a period of 6 s. The system is turned on at 0 in buck converter mode. Initially strong irregularities arise, until steady-state operation is attained. Fig. 13 shows the power losses of the first chip working as low side freewheeling diode for the same module operation mode with considerably higher peak values of dissipated power compared to the high side switches. Also after time integration the diode losses are higher and cause a considerable higher self heating compared to the high side switches, as can be seen from Fig. 14. The switching instances in Fig. 14 can clearly be read
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Fig. 16. Same as in Fig. 14, but converter frequency reduced to 100 kHz (10-s period). Fig. 13. Same as in Fig. 12, however, losses of first chip working as low side freewheeling diode.
Fig. 17. Converter short circuit operation High side switch permanently turned on. Low side diode is in off-state. Fig. 14. Simulation result for the temperature evolution of the two chips of Figs. 12 and 13. Converter frequency 170 kHz.
Fig. 15. Simulation time of Fig. 14 extended to 4 ms with 8000 MOSFET/diode turn-off/on switchings for the six-chip module.
off from the temperature evolution. The on-state losses between turn-on and turn-off correspond to an average device current of 50 A. The turn on/off pulses last for approximately 200 ns. Fig. 15 shows the temperature evolution with simulation time extended to 4 ms and altogether 8,000 MOSFET/diode turnoff/on switchings for the six-chip module. This takes 1-h computation time on a fast PC. The plot resolution can no longer resolve the temperature oscillations as in Fig. 14. In Fig. 16, the converter frequency is reduced to 100 kHz (10- s period) compared to 170 kHz in Fig. 14 which reduces at most the temperature increase of the high side switch. In this case the inductance in Fig. 10 has been increased from 2 H to 3.3 H for all three parallel dc/dc-converters in order to reduce the current variation. Because of the small temperature oscillations in the heating curves of Fig. 14–16, a simplified calculation with some suitable averaging of the input power loss could be used. However, for unsteady converter operation with large temperature sweeps the
full electrothermal simulation is superior. Fig. 17 displays the temperature evolution in the MOSFETs for a short circuit condition at the output side or for a load with impedance reduced from 0.1 to 0.05 . In order to maintain a constant output voltage the converter works with its highest duty ratio , i.e., the high side switches are nearly constantly turned on and the low side switches turned off. The losses of the low side switch in its blocking state are negligible. Nevertheless its temperature rise is noticeable (14 C in 2 s) due to the dissipated power in the neighboring high side switches. This is an essential effect of the thermal network part of the electrothermal model which accounts for the thermal interaction of the different chips. VI. CONCLUSION A complete thermal model was derived with linear, but otherwise general time-dependent bound.c., starting from the eigenfunction representation of the Green’s function without calculating the eigenfunctions themselves. With the help of a small set of 20 effective time constants this lead to a temperature field representation with low order matrices , including the effects of varying surface heat sources and ambient temperatures applied at thermal contact areas. By this “separation” of variables , the 3-D-field at any time can be calculated very fast (in less than 100 s) for an pulse width modulated automotive dc/ac converter with inclusion of varying ambient . The model was obtained by linear fits to FEM simulated matrix heating curves (thermal impedances). For a reduced number of hot spot locations the thermal model leads directly to a transient, reduced thermal model. In addition a rigorous and general method to construct reduced transient thermal equivalent circuits for the compact models results with arbitrary initial conditions of the temperature field. The thermal models and equivalent circuits also allow for temperature-dependent volume and
GERSTENMAIER et al.: ELECTROTHERMAL SIMULATION OF MULTICHIP-MODULES
surface heat sources, so that a full electrothermal simulation is possible. Combining the thermal model with an electrical PSpice MOSFET model electrothermal simulations of multichip modules over thousands of converter cycles can be performed with inclusion of the thermal interaction of the chips. Thus a useful tool for the optimization of the system design is established.
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By inserting this in the expression (11) for
(13) and using (13) leads to the following expression for the volume integral over the heat sources , the second term of (3)
APPENDIX A. Thermal Model Equations With the help of the eigenfunctions and corresponding of the spatial differential operator of the heat eigenvalues conduction (1) or (14) the Green’s function [2], [17]
occuring in (3) can be represented by with (11)
denotes the unit step function. The positive eigenwhere value spectrum of this equation depends on the bound.c. are subjected to. are the time conthe eigenfunctions stants of the problem [6], [20]. or For general system geometries, the eigenfunctions the Green’s functions are difficult to calculate. In [27], [28] the Green’s function method has been used for rectangular one- and two-layer systems, which, by a set of effective parameters determined by measurement or simulation, describe more complex multilayer structures. The eigenfunctions and time conhave been calculated in [6] for multilayer structures stants similar to Figs. 1 and 2, however, all layers had equal lateral extensions in order to make possible a semi-analytical calculaform a complete system of functions with hotion. The , mogeneous or “zero” bound.c.: so that the solution of (1) with these boundary conditions can . It is important to be expressed as note, that the are not orthogonal in the customary sense, but only with respect to a special inner product defined as [6]: . The heat generation density may include several independent heat sources (chips in a MCM) with dissipating power , 1 , with describing the distribution of the heat generation in chip , 1. can be represented by The infinite set of time constants by a small set of approximating the exponentials logarithmically distributed effective time constants [1], [23] which have been introduced following (7) with
(12)
and
For electrothermal simulation the may also depend on . Equation the chip-temperatures (14) then becomes an implicit nonlinear equation for . matrix constitutes the thermal model of the set-up The which is valid for temperature fields with homogeneous or “zero” bound.c. In order to complete the model for general bound.c., averaged at thermal contact areas 1 are temperatures 0 (and considered. In case of pure Dirichlet condition with 0) on that parts of , as discussed following (3), the surface integral of (3) leads to the contribution for the applied with 1 bound.c. (15) with
and
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006
When variable ambient temperature and a heat transfer are present instead of a Dirichlet condition at , coefficient the surface integral of (3) has to be used with integrand (5) and which gives
denotes the distribution of the ambient temperature at . contact with averaged temperature At last for pure heat flux bound.c. (i.e., (5) with 0) along thermal contact the surface term of (3) again leads for with heat flux distribution 1 to (16)
for and . The weighting factor for 0 gives more weight to small time values in the fitting process. A value 1 Average is appropriate, but also for 0 of good results are obtained. fits instead of monotThe program for determining monotonously deonously increasing functions creasing functions which of are approximated by linear combinations ) with the subsidiary the basis functions . The condition are obtained by minimizing the norm which leads to . The matrix is easily calculated. The additional leads to an equation system with order condition reduced by one by eliminating
inner product
with (18)
and
where is the heat flow at thermal contact . The complete temperature field is the sum of (14)–(16), which gives rise to the expression (6) for . B. Algorithm for Least Square Fit , , and To determine the model matrices in (6) FEM simulated heating curves (thermal impedances or ) for the different single source unit step responses heating conditions are established from which the model parameters are deduced by linear least square fit. with Using the expression (7) for the temperature field and 0 for all the temperature 0 is field for homogeneous starting temperature (17) denotes the unit step response of the system for heating only, e.g., chip with unit strength of power or one thermal contact area with a unit temperature step. is the generalized thermal impedance for fixed position . It can serve to calculate for arbitrary time evolution by with the time derivative [20], convolution of [29], which can also be directly inferred from (3). are the coefficients of According to (17) the basis functions , caldirectly by linear least square fits culated by adjusting without calto FEM-simulated impedances , or . As has been pointed out in [1] culating the the least square fit can be done with the help of an integral defined by means of the norm
The ( , ) elements are obtained by linear interpolation of the from FEM-simulation and subsequent andiscrete values ). Thus only alytical integration in the time intervals ( , very rapidly calculated analytical elements enter the governing (18) for the unknowns . The matrix inversion of has only to be done once and a very fast determination for all grid points of the FEM model results. of the The ordinary least square fit method using only sampling without integral norm works well—but not quite points as reliable—for time sampling points distributed densely for small time values and having increasing distance for large times. The inclusion of the subsidiary condition for the in both cases leads to a reduced linear system of degree 1.
REFERENCES [1] Y. C. Gerstenmaier and G. Wachutka, “Efficient calculation of transient temperature fields responding to fast changing heatsources over long duration in power electronic systems,” IEEE Trans. Comp. Packag. Technol., vol. 27, no. 1, pp. 104–111, Mar. 2004. [2] , “Efficient calculation of transient temperature fields with general boundary conditions in electronic systems,” in Proc. 10th MIXDES Conf., Lodz, Poland, Jun. 2003, pp. 313–318. [3] Ansys, Inc. (2005). Tech. Rep. [Online] Available: http//www.ansys.com [4] V. Székely and M. Rencz, “Fast field solvers for thermal and electrostatic analysis,” in Proc. Design, Automation Test Europe, Paris, France, Feb. 1998, pp. 518–523. [5] A. Csendes, V. Székely, and M. Rencz, “An efficient thermal simulation tool for ICs, microsystem elements and MCMs: the s-THERMANAL,” Microelectron. J., vol. 29, pp. 241–255, 1998. [6] Y. C. Gerstenmaier and G. Wachutka, “Time dependent temperature fields calculated using Eigenfunctions and Eigenvalues of the heat conduction equation,” Microelectron. J., vol. 32, pp. 801–808, 2001. [7] C. J. M. Lasance, D. den Hertog, and P. Stehouwer, “Creation and evaluation of compact models for thermal characterization using dedicated optimization software,” in Proc. IEEE SEMI-THERM XV, San Diego, CA, 1999, pp. 189–200. [8] H. Rosten, C. J. M. Lasance, and J. Parry, ““The world of thermal characterization according to DELPHI—part I: background to DELPHI” and “part II: experimental and numerical methods”,” IEEE Trans. Comp., Packag., Manufact. Technol. A, vol. 20, no. 4, pp. 384–398, Dec. 1997. [9] M. N. Sabry, “Static and dynamic thermal modeling of ICs,” Microelectron. J., vol. 30, pp. 1085–1091, 1999.
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[10] H. Pape and G. Noebauer, “Generation and verification of boundary independent compact thermal models for active components according to the DELPHI/SEED methods,” in Proc. IEEE SEMI-THERM XV, San Diego, CA, 1999, pp. 201–207. [11] Y. C. Gerstenmaier, H. Pape, and G. Wachutka, “Rigorous model and network for static thermal problems,” Microelectron. J., vol. 33, pp. 711–718, 2002. [12] F. Christiaens, B. Vandevelde, E. Beyne, R. Mertens, and J. Berghmans, “A generic methodology for deriving compact dynamic thermal models, applied to the PSGA package,” IEEE Trans. Comp., Packag., Manufact. Technol. A, vol. 21, no. 4, pp. 565–576, Dec. 1998. [13] M. Rencz and V. Székely, “Dynamic thermal multiport modeling of IC packages,” IEEE Trans. Comp. Packag. Technol., vol. 24, no. 4, pp. 596–604, Dec. 2001. [14] Y. C. Gerstenmaier and G. Wachutka, “Rigorous model and network for transient thermal problems,” Microelectron. J., vol. 33, pp. 719–725, 2002. [15] D. Schweitzer and H. Pape, “Boundary condition independent dynamic thermal compact models of IC-packages,” in Proc. 9th THERMINIC, Aix-en-Provence, France, Sep. 2003, pp. 225–230. [16] W. Batty, C. Christofferson, A. J. Panks, S. David, C. M. Snowden, and M. B. Steer, “Electrothermal CAD of power devices and circuits with fully physical time-dependent compact thermal modeling of complex nonlinear 3-D systems,” IEEE Trans. Comp. Packag. Technol., vol. 24, no. 4, pp. 566–590, Dec. 2001. [17] Y. C. Gerstenmaier and G. Wachutka, “Transient temperature fields with general nonlinear boundary conditions in electronic systems,” IEEE Trans. Comp. Packag. Technol., vol. 28, no. 1, pp. 23–33, Mar. 2005. [18] M. Rencz and V. Székely, “Studies on the nonlinearity effects in dynamic compact model generation of packages,” IEEE Trans. Comp. Packag. Technol., vol. 27, no. 1, pp. 124–130, Mar. 2004. [19] J. V. Beck, K. D. Cole, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction Using Green’s Functions. Bristol, PA: Hemisphere, 1992. [20] Y. C. Gerstenmaier and G. Wachutka, “Calculation of the temperature development in electronic systems by convolution integrals,” in Proc. IEEE SEMI-THERM XVI, San Jose, CA, 2000, pp. 50–59. [21] M. B. Gerber, “On packaging techniques for a high power density dc/dc converter,” M.S. thesis, Rand Afrikaans Univ., Johannesburg, South Africa, Dec. 2001. [22] N. Seliger, J. Rackles, H. Schwarzbauer, W. Kiffe, and S. Bolz, “Compact and robust power electronics packaging and interconnection technology,” in Optimization of the Power Train in Vehicles by Using the Integrated Starter Generator (ISG), H. Schäfer, Ed. Munich, Germany: Verlag, 2002, pp. 248–255. [23] Y. C. Gerstenmaier and G. Wachutka, “Compact thermal model for transient temperature fields in electronic systems,” in Proc. 5th Conf. Modeling Simulation Microsystems (MSM’02), San Juan, Puerto Rico, Apr. 2002, pp. 608–611. [24] Eupec, Inc. (2005). Simulation tool IPOSIM. [Online] Available: http://www.eupec.com [25] Infinion. (2005). [Online] Available: http://www.infineon.com [26] Microsim, PSpice User Manual, 2005. [27] M. Janicki, G. DeMey, and A. Napieralski, “Application of Green’s functions for analysis of transient thermal states in electronic circuits,” Microelectron. J., vol. 33, pp. 733–738, 2002. [28] M. Janicki and A. Napieralski, “Analytical transient solution of heat equation with variable heat transfer coefficient,” in Proc. 8th THERMINIC, Madrid, Spain, Oct. 2002, pp. 235–240. [29] Y. C. Gerstenmaier and G. Wachutka. A new procedure for the calculation of the temperature development in electronic systems. presented at EPE’99 Conf. [CD-ROM]
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York C. Gerstenmaier (M’90) received the M.S. and Ph.D. degrees in physics from the University of Bonn, Bonn, Germany. His doctoral research was in the area of quantum mechanical many-body theory. In 1987, he joined the Power Electronics Department, Siemens Corporate Technology, Munich, Germany. Since then he has been involved in the development of power electronic components and systems, modeling tools, and simulation. He has authored more than 40 scientific papers. Currently, he is concerned with thermal and electrothermal modeling and development of power electronic systems. Dr. Gerstenmaier is a member of the THERMINIC Programme Committee.
Alberto Castellazzi (M’04) was born in Treviglio, Italy, in 1972. He received the M.S. degree in physics from the University of Milan, Italy, in 1998 and the Ph.D. degree in electrical engineering from Munich University of Technology, Munich, Germany, in 2004. From 1998 to 2000, he was with Carlo Gavazzi Space, Milan, as a Designer of power electronics for space applications. In 2000, he joined the Power Electronics Department, Siemens Corporate Technology, Munich, studying the reliability of power MOSFETs in novel automotive applications. He is a Research Scientist at the Institute for Physics of Electrotechnology, Munich University of Technology. His research interests are dc–dc converters, electrothermal device simulation, and investigation of power device reliability. Dr. Castellazzi is a member of the IEEE Power Electronics Society, IEEE Electron Devices Society, and the European Power Electronics Association.
Gerhard K. M. Wachutka received the D.Sc. degree from the Ludwig-Maximilians-Universität, Munich, Germany, in 1985. From 1985 to 1988, he was with Siemens Corporate Research and Development, Munich, where he headed a modeling group active in the development of modern high-power semiconductor devices. In 1989, he joined the Fritz-Haber-Institute of the Max-Planck-Society, Berlin, Germany, where he worked in the field of theoretical solid-state physics. From 1990 to 1994, he was head of the microtransducers modeling and characterization group of the Physical Electronics Laboratory, Swiss Federal Institute of Technology (ETH), Zürich, Switzerland. He has authored or coauthored more than 180 publications in scientific or technical journals. He is a consultant with various research institutes in industry and academia. Among his many educational activities, he has set up and taught courses funded by European Community training programs. Since 1994, he has been head of the Institute for Physics of Electrotechnology, Munich University of Technology, where his research activities are focussed on the design, modeling, characterization, and diagnosis of the fabrication and operation of semiconductor microdevices and microsystems. Dr. Wachutka is member of the American Electrochemical Society, the American Materials Research Society, the ESD Association, the VDE Association for Electrical, Electronic and Information Technologies, the German Physical Society, the American Physical Society, and the AMA Society for Sensorics.
