A method for tightly coupled thermal-electrical simulation Bernhard Klaassen GMD-SET Institute D-53754 St. Augustin, Germany e-mail:
[email protected] Tel: ++49-2241-142851, fax: ++49-2241-142242
Abstract A method is presented to combine analog simulators (like SPICE) with device simulators (like ANSYS), for thermal-electrical analysis of integrated circuits or systems. A sketch of a rst prototype program is given together with the theoretical background, which makes use of convergence principles from Waveform Relaxation. It follows a short review of related work. Our approach can also be extended to more general problems occuring in microsystems simulation.
1 Introduction The designer of integrated circuits and systems (as well as the manufacturer) needs reliable simulation methods that also take into account the system's environment. Prede ned standard conditions (e. g. constant overall temperature) are no longer a reasonable assumption, since the circuit's electrical behaviour in uences the temperature and vice versa. This can be important for the behaviour of the whole system. Other physical e ects, like electro-magnetic elds, are typical for many sorts of strong or weak couplings between components of the designed system. Considering the increase of e orts to integrate not only electronics but as well micromechanical, optical, and other parts on the same chip, one can predict that the demand for these simulation methods will increase, too. On the other hand it is no longer true that solvers for partial di erential equations (PDE) need a most powerful `number cruncher'. Even with a good workstation one can get PDE-results within reasonable time. If one takes into account that several mechanical e ects also need a PDE-modelling (when high accuracy is needed), the additional e ort for a thermal PDE-analysis of the entire chip seems to be justi able. This paper presents a method to combine solvers for ODEs (ordinary di erential equations) and PDEs like e. g. SPICE [5][7] and ANSYS(TM) [2].
2 An example To illustrate our problem we shall give a sketch of a practical example. Let us consider a sensor for acceleration built into a wheel of a motorcar together with an electronic control system to prevent locking. Normally piezo-resistive elements are used for the sensor. Since these are very temperature sensitive elements the in uence of heating between system parts cannot be ignored. There are several types of accelerometers on silicon. One usual technique is to x a piece of metal as a weight on a bendable silicon tongue, whose deformation can be detected by a set of
four piezo-resistive elements building a Wheatstone bridge. We need some more electronic parts for A/D-conversion, signal evaluation, etc. on the same chip or nearby. To model this system including its temperature behaviour we have to consider the following levels: PDE thermal simulation of system
ODE
electrical simulation Forces PDE electric-mechanical simulation (Piezo)
Fig. 1: Levels of simulation for mixed system Obviously each level of simulation in uences the two other ones directly or indirectly. One has to keep in mind that each level of simulation represents a transient i.e. a time varying behaviour with largely di ering time constants. Therefore it is sometimes possible to decouple the levels of simulation if the in uence of temperature on the two lower levels is not strong. But when sensors of the described kind are involved, this simpli cation will cause errors. To eliminate these errors we introduce an iterative technique using time-slices (so called \windows"): 1. Chose a window according to the biggest actual time step from all levels. 2. Compute each level within the window starting with bottom level using actual values (or inter/extrapolation in time) from other levels. 3. Continue step 2 until di erence to previous iteration is below a given tolerance. Then proceed with step 1 until end of simulation interval. One should note that within each window the di erent solvers on each level are allowed to choose their own appropriate step sizes. This property is called \multirate" e ect in numerical literature. For our application, where the size of time steps can di er by a factor 100 and more between the simulation levels, it is absolutely necessary to develop multirate methods to keep CPU-time amounts within reasonable bounds.
3 Theoretical background The mathematical model for the top level of Fig. 1 is the well known heat equation for temperature at each point within the chip:
c
( ) = +
@ t @t
X( e ( ) eE
v t
( ) vole )
ie t =
(1)
with initial temperature = 0 at time t0 and appropriate boundary conditions for convection etc. With: material density c speci c heat thermal conductivity (possibly direction dependent) Laplace operator (sum of second partial space deriv.) E set of circuit elements ve voltage at element eE ie current through element eE vole volume of element eE The terms under the sum are position dependent and their value is zero outside of the electrical circuit elements. They can be considered as heat sources within certain volume parts. Let us now apply a standard discretization technique (see e. g. [1]) of ` nite di erences' resulting in a space grid. At each grid point we nd a di erence equation derived from (1) where the space derivatives of the -operator are replaced by weighted mean values. It can be assumed that this grid has been chosen ne enough to yield the desired accuracy (there can also be local re nements). Since time derivatives remain unchanged by this process, we yield one ODE for the temperature j at each of the k grid points. If we write k for the k-dimensional vector of all j , the resulting system of all these equation has the form k _k (t) = g (t; (t); y (t)); (2) where g() comes from the right side of (1) discretized and divided by c, and y represents the vector of unknown currents and voltages, which must be calculated by the usual circuit equation C (t; y; k(t))y_ (t) = f (t; y (t); k (t)): (3) Equations (2) and (3) together form a system of ODEs for k and y. From the theory of relaxation methods [10] it is known that such a system can be solved iteratively by decoupling the parts (2) and (3) and computing in an alternating way (analogue to the Waveform Relaxation method with only two `groups'). At each iteration the actual solution from one part must be transmitted as input into the other one and vice versa. If convergence becomes slow (e. g. more than 5 iterations) the chosen time window can be reduced. This must increase the speed of convergence but limits the multirate e ect. For more detailed investigations of this eld see also [4], where this approach has been published rst (in German).
4 Realization In the previous section the method of nite di erences was mentioned for numerical solution of the PDE parts in our example system. The general convergence properties of the iterative process allow some freedom in the choice of the particular solvers. Since the ODE part doesn't make use of the actual PDE solving method (it only needs approximated solution values) we are free to select the ` nite element' method (FEM) instead of nite di erences. The practical reason for this decision was the wider propagation of FEM in this area of application. Since ANSYS (as well as SPICE) has many users, we decided to build a prototype realization of our proposed method with these two packages. (Nevertheless it is obvious that these interface and control parts can easily be rebuilt for many other combinations of simulation tools.) We use the following structure using SPICE3 and ANSYS5.0:
Preprocessing Phase: Build a mapping from the space coordinates of the SPICE elements
(from the layout) into the corresponding volume elements in ANSYS. (It need not be a one-toone mapping, since the desired accuracy may allow a coarse FEM meshing.1 ) Using this mapping build an ANSYS command le which lists the SPICE elements as heat generating sources within the corresponding ANSYS volume elements. Build a list of ANSYS nodes for output of nodal temperature values, from which the corresponding SPICE element temperatures can be interpolated using a SPICE command le.
Iteration start: Chose window size. Start ANSYS piezo simulation with given forces and
constant temperature. Perform SPICE iteration with updated piezo-resistance and constant temperature. Update heat generation rates in the ANSYS command le. ANSYS waits for this le and starts thermal iteration with actual values. Output is the SPICE command le with new temperature values.
Iteration phase: Each simulator waits for his new command le to perform one iteration
within the window. Actual results are compared to the previous ones and iteration stops if results are stable within given tolerances (e. g. 0.1V and 0.5 degrees). Then `iteration start' begins with new time window.
The following picture (Fig. 2) shows results of thermal analysis of a presure sensor chip with some opamps in the upper left region ( max = 52o C; min = 27o C ):
MX
MN Fig.2
Fig. 2: Thermal pro le of sensor chip
It may be possible that a SPICE element belongs to more than one ANSYS volume elements. Then the corresponding heat source could be partitioned but in most cases it is easier (and accurate enough) to chose one central volume element to contain the heat generation of the SPICE element. 1
The e ect of heating on the sensor output is demonstrated clearly in the next gure: V
with temp
without temp
15. 14. 13. 12. 11. 10. 9. 8. 7. 0.
20.
40.
60. time
80.
100. 120. mS
140.
160.
Fig. 3: Sensor output with/without temperature simulation
The somewhat crinkly look of the upper curve is not an accuracy problem but comes from a simple implementation problem in SPICE: It was not possible to introduce a time-dependent temperature value. So the only way was to use temperature as a stair function changing at each time window. For our prototypical implementation this was enough but in a more professional environment this means: The electrical simulator must allow changing temperature (resp. must be tuned to allow this).
5 Related work In the eld of electrical simulation some alternative approaches exist to model heating e ects together with the system of network equations. In particular we can distinguish three di erent lines: A) \Model transfer": Transform PDEs into ODEs. B) \Model coupling": New type of simulator. C) \Simulator coupling": E. g. this paper. About A): One of these strategies is that additional resistors are inserted into each transistor model adjusting their conductance values proportionally to the assumed thermal conductivity (see e. g. [8]). This technique is limited w.r.t. accuracy, since it cannot deliver a complete temperature eld. It is also necessary to modify each new device model in the described manner, which becomes more complicated when we consider mechatronics and other new areas where models are still to be developed. A more general approach is [6], where PDEs are transformed automatically into sets of equations suitable for an analog simulator like SPICE. About B): There exist rst realizations of analog simulators (like e. g. SIMULise from ise, Zurich) which incorporate modelling of both ODE and PDE parts. In general this is a `natural' evolution since the user will often prefer a tool which combines the capabilities of two existing ones. A drawback might be the fact that many users work as specialists with their special simulators and have no desire to throw their modelling experience away for a new simulator. About C): The main disadvantage of simulator coupling is, of course, the additional overhead
for communication and iteration. While the algorithmical side has already been treated in this paper, there are some contributions still to be mentioned to overcome this drawback. As an example for a communication concept serves e. g. [9]. One main advantage of simulator coupling in contrast to B) or A) is the fact that the modelling for the two (or more) di erent simulator sides can be left as it is. This can often save more time than the whole simulation needs for computation. Another big advantage is concerning runtime: All known approaches result in a system of timedependent equations, all to be solved on the same time discretisation. In the case of thermal simulation this means that e. g. electrical signals moving much faster than the temperature would dominate the timesteps of the whole simulation process. This e ect will not occur in our iterative coupling approach, as long as the time window is chosen in an appropriate manner. A simple estimation shows that this advantage is able to easily compensate the drawback of iteration overhead.
6 Conclusion and future work A method (and a prototype software) have been presented for tightly coupled thermal-electrical analysis of circuits and microsystems. There exist several commercial design systems which compute a temperature eld over a chip, but they usually do not take into account the feedback loop between power dissipation of electrical parts and the temperature dependent behaviour of nonelectric and electronic components. First numerical results with our method indicate that especially this thermal `feedback' cannot be ignored when reliable simulation results are needed. Two widespread simulators (SPICE3 and ANSYS5.0) are used, but the fact should be stressed that the coupling method is suitable for any other pair of simulators which use a programmable shell or any other similar command le interface. The method makes use of several modelling levels from device equations (PDE) through circuit equations (ODE) to global heat equation (PDE). This concept of levels with iterative solution process could easily be extended to include the system environment. If e. g. in our rst example (\car wheel") also the mechanical equations for the wheel's motion were to be included, we should have to insert a top level of ODEs (see Fig. 4).
ODE
mechanical simulation of environment
PDE thermal simulation of system (plus environment)
ODE
electrical simulation (integrated circuits)
PDE electric-mechanical simulation (eg Piezo)
Fig. 4: Enhanced levels of coupled simulation
For such a more general problem of couplings also the mathematical methods have to be generalized. Together with our Russian collegue V. B. Dmitriev-Zdorov we were able to realize such an appraoch for a simulation of an ultrasonar control device. (See [3], where several types of coupling are discussed). The main idea of that approach was to improve the convergence properties by introducing `rough' models for the behaviour of the nonelectrical parts. Fortunately, for the temperature case, the relatively easy type of coupling which has been described here, is completely sucient.
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[6] G. Pelz, J. Bielefeld, F.-J. Zappe, G. Zimmer, \MEXEL: Simulation of Microsystems in a Circuit Simulator Using Automatic Electromechanical Modeling", in: H. Reichl, A. Heuberger (Edts.) \MICRO SYSTEM Technologies 94", 4th International Conference on Micro Electro, Opto, Mechanic Systems and Components, vde-verlag, Berlin, 1994. [7] T.L. Quarles, \Analysis of performance and convergence issues for circuit simulation", Dissertation, Memo UCB/ERL M89/42, Berkeley, CA., 1989. [8] H. Rudolph, D. Seitzer, \Thermal Analysis of Integrated Circuits", in: H. Reichl (Edt.) \MICRO SYSTEM Technologies 90", 1st International Conference on Micro Electro, Opto, Mechanic Systems and Components, Springer-Verlag, Berlin, 1990. [9] S. Schmerler, Y. Tanurhan, K.D. Muller-Glaser, \A Backplane Approach for Cosimulation in High-Level System Speci cation Environments", Proc. of EURO-DAC'95, IEEE Comp. Soc. Press, 1995. [10] J. White, A.L. SangiovanniVincentelli, \Relaxation Techniques for the Simulation of VLSI Circuits", Kluwer Academic Publishers, Boston, 1986.
