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lg HZStep1 @KêWD L0-0.5-1-1.5-2-4 -3 -2 -1 0 1 2 3lg HTime @secDLFig. 10: Unit step response ZStep1(t) is time integrated impulse responseZ1(t), which in turn is inverse Laplace transform of Z1(s) in Fig.9 (12).Z 2 connected to ground in Fig.9, physically reasonable functionsZStep 1, 2 (t) are obtained by the procedure and the correspondingFoster R i , C i can be read of from the Z 1, 2 (s)when they are composed in partial fractions. The ZStep 1 (t)curve is shown in Fig.10 as double logarithmic plot. TheZStep 2 (t) curve shows only minor deviations from ZStep 1 (t).The Z 1, 2 (s) always had poles only on the negative s-axesand no complex poles. For poor approximation of the originaldata by model (2) in the time domain this is no longertrue. The real problem, however, is the transfer impedanceZ 3 (s) between the heat sources. For the example under considerationthis function always revealed completely unphysicalbehaviour, e.g. by poles on the positive axes with correspondingexponential increase in the time domain or by oscillations.Also in cases with stable circuit impedance nophysical behaviour of ZStep 3 (t) could be obtained.This failure might be related to the approximation methodfor the impedance matrix (2), which is accurate in time domainbut not necessarily so in the frequency domain, whereall impedances are represented by Foster RC-chains (4). Ashas however been pointed out in section IV, some impedancelinks in the network model (Fig.6, 8) may better be representedby polynomials in s.Therefore it seems to be generally not advisable to proceedfor the determination of the network parameters in theway indicated here. A better way would be to determine thenetwork admittances y ik (t) directly in the time domain bymeasurement or simulation as suggested at the end of sectionIII. A one port link y ik (t) can be fitted by a product of a polynomialin t with a linear combination of exponentials in t.This function can exactly be transformed in Laplace domainfor the determination of the corresponding R, C, L circuitelements. Further investigations have to be done.CONCLUSIONSThe suggested new network model, which uses only oneportimpedance links, allows for the determination of both:the transient temperatures at specified thermal contacts andthe associated heat flows at the contact areas. The networkhas a considerable advantage in determining model- parameterscompared to previous models with Cauer three terminallinks. When m is the number of thermal device contact areasand p the number of heat sources, the model is characterizedby (m + p +1) (m + p) / 2 one-port impedances with its associatedR, C, L elements. One presented methodology for the24-26 September 2008, Rome, Italydetermination of the network parameters poses in manycases a highly ill conditioned problem, which may render theresults useless. As alternative method direct measurement/simulation of the model link admittances y ik (t) in the timedomain is recommended for determining the parameters.REFERENCES[1] M.N. Sabry, “Dynamic Compact Thermal Models”, lecture presented atTHERMINIC, Madrid, Spain, Oct. 2002.[2] C.J.M. Lasance, D. den Hertog, and P. Stehouwer, “Creation andEvaluation of Compact Models for Thermal Characterization UsingDedicated Optimization Software,” in Proc. IEEE SEMI-THERM XV,San Diego, USA, 1999, pp.189-200.[3] H. Rosten, C.J.M. Lasance, and J. Parry, “The world of thermal characterizationaccording to DELPHI- Part I: Background to DELPHI” and“Part II: Experimental and Numerical Methods,” IEEE Trans. Comp.,Hybrids, Manufact. Technol., vol. 20, pp.384-398, Dec. 1997.[4] M.N. Sabry, “Static and dynamic thermal modeling of ICs,” MicroelectronicJ. vol. 30, pp.1085-1091, 1999.[5] H. Pape and G. Noebauer, “Generation and verification of boundary independentcompact thermal models for active components according tothe DELPHI / SEED methods,” in Proc. IEEE SEMI-THERM XV, SanDiego, CA, 1999, pp.201-207.[6] Y.C. Gerstenmaier, H. Pape, and G. Wachutka, “Rigorous model andnetwork for static thermal problems,” Microelectronic J., vol. 33,pp.711-718, 2002.[7] F. Christiaens, B. Vandevelde, E. Beyne, R. Mertens, and J. Berghmans,“A Generic Methodology for Deriving Compact Dynamic ThermalModels, Applied to the PSGA package,” IEEE Trans. on Components,Packaging and Manufacturing Technology, Part A Vol 21, No.4, pp 565-576, 1998.[8] M. Rencz and V. Székely, “Dynamic thermal multiport modeling of ICpackages,” IEEE Trans. on Comp. Packag. Technol., vol 24, No.4, pp596-604, 2001.[9] Y.C. Gerstenmaier and G. Wachutka, “Rigorous model and network fortransient thermal problems,” Microelectronic J., vol. 33, pp.719-725,2002.[10] D. Schweitzer and H. Pape, “Boundary Condition Independ-ent DynamicThermal Compact Models of IC-Packages,” Proc. 9thTHERMINIC, Aix-en-Provence, France, Sept. 2003, pp.225-230.[11] W. Batty , C. Christofferson, A.J. Panks, S. David, C.M. Snowden, andM.B. Steer, “Electrothermal CAD of Power De-vices and Circuits withFully Physical Time-Dependent Compact Thermal Modeling of ComplexNonlinear 3-D Systems,” IEEE Trans. Comp. Packag. Technol.,Vol.24, No.4, pp. 566-590, 2001.[12] L. Codecasa, D. D’Amore, P. Maffezzoni, “Compact Thermal Networksfor Modeling Packages,” IEEE Trans. Comp. Packag. Technol., vol.27,pp.96-103, Mar. 2004.[13] Y.C. Gerstenmaier, G. Wachutka, “Efficient Calculation of TransientTemperature Fields Responding to Fast Changing Heat-sources OverLong Duration in Power Electronic Systems,” IEEE Trans. Comp.Packag. Technol., vol.27, pp.104-111, Mar. 2004.[14] Y.C. Gerstenmaier, A. Castellazzi, G. Wachutka, “Electro-thermalSimulation of Multichip-Modules with Novel Transient Thermal Modeland Time-Dependent Boundary Conditions,” IEEE Trans. Power Electronics,vol.21, pp.45-55, Jan. 2006.[15] M.N. Sabry, “Effect of heat flux patterns on the precision of compactthermal models,” Proc. 10th THERMINIC, Sophia Antipolis, France,Oct. 2004.[16] V. Székely, “Identification of RC networks by deconvolution: chancesand limits,” IEEE Trans. Circuits Syst.-I, vol. CAS-45, pp.244-258,March 1998.[17] Y.C. Gerstenmaier, W. Kiffe, G. Wachutka, “Combination of thermalsubsystems by use of rapid circuit transformation and extended two-porttheory,” submitted to Microelectronic J..[18] L. Weinberg, Network Analysis and Synthesis, New York, McGraw-Hill,1962.[19] Wai-Kai Chen, The Circuits and Filters Handbook, chapters 73 – 76,2nd edition, CRC Press, 2003.©EDA Publishing/THERMINIC 2008 75ISBN: 978-2-35500-008-9