Budapest, Hungary, September 2007 Combination of Thermal Subsystems Modeled by Rapid Circuit Transformation

Transcription

1 Budapes, Hungary, 7-9 Sepember 27 Combinaion of Thermal Subsysems Modeled by Rapid Circui Transformaion Y.C. Gersenmaier*, W. Kiffe*, and G. Wachuka** *Siemens AG, Corporae Technology, 873 Muenchen, Germany, **Insiue for Physics of Elecroechnology, Munich Universiy of Technology, Germany Absrac- This paper will deal wih he modeling-problem of combining hermal subsysems (e.g. a semiconducor module or package wih a cooling radiaor) making use of reduced models. The subsysem models consis of a se of Foser-ype hermal equivalen circuis, which are only behavioral models. A fas algorihm is presened for ransforming he Foser-ype circuis in Cauer-circuis which have physical behavior and herefore allow for he consrucion of he hermal model of he complee sysem. Then he se of Cauer-circuis for he complee sysem is ransformed back ino Foser-circuis o give a simple mahemaical represenaion and applicabiliy. The ransformaion algorihms are derived in concise form by use of recursive relaions. The mehod is exemplified by modeling and measuremens on a single chip IGBT package mouned on a closed waer cooled radiaor. The hermal impedance of he complee sysem is consruced from he impedances of he subsysems, IGBT-package and radiaor, and also he impedance of he package can be inferred from he measured impedance of he complee sysem. I. INTRODUCTION For he calculaion of ho spo emperaures or emperaure fields in elecronic sysems wih rapidly varying chip hea source srengh usually reduced models are used. Numerous compac saic [-5] and ransien [6-] hermal models have been esablished for a rapid calculaion of emperaures. The noion of compac hermal model usually implies boundary condiion independence (BCI) [, 2], i.e. he model is valid for all (or nearly all) reasonable emperaures, hea flows and also hea ransfer coefficiens applied o he hermal conac areas. An advanage of he model presened in [, 2] is is ease of parameer deerminaion by simple linear leas square fi o measured or simulaed heaing curves (hermal impedance. The model was exended in [2] o include he effecs of varying surface or ambien emperaure and varying hea flows a he hermal conac areas. However, i is no possible o use arbirary hea ransfer coefficiens α as exernal bound.c. parameers independenly of he model parameers. Thus new model parameers have o be deermined, when α changes. On he oher hand he compac models of [, 2, 4, 5] deal wih small packages wih small hermal conac areas in comparison o he muli chip module of Fig., which is mouned on a cooling radiaor. Such modules always have a large emperaure variaion along he boom side of he module base plae depending on he differen heaing cases. Thus he module can hardly be approximaed hermally by a compac model which is independen of α. MN Mos of he reduced models, also [, 2], apply o sysems wih negligible nonlineariies, e.g. he maerial parameers are supposed o be emperaure independen jus as is α. Mehods for reducing nonlinear sysems are presened e.g. in [3, 4]. This paper will deal wih he modeling-problem of combining hermal subsysems, e.g. combining a power semiconducor module wih a cooling radiaor, making use of he reduced models of [, 2]. Hea ransfer coefficiens α in his conex can also be modeled as hermal subsysem (layer of hermal resisance). II. THERMAL MODEL The model in [, 2] characerizes he hermal se-up by a se of M 2 effecive ime consans i which are logarihmically disribued, ypically beween Min( i ) = -4 s and Max( i ) = s (depending on sysem and hea source size), and besides his are chosen freely. The model equaion for he emperaure field T(x, reads: M L+ C+ J il i= l= MX X Fig. : FEM-simulaion of seady sae emperaure ( K) disribuion for 7V IGBT and diode module mouned on cooling radiaor. τ T( x, = M ( x) s ( τ ) e i dτ () where s l ( denoes a hea source erm which can be eiher he dissipaed power p l ( of a chip l, l =,.., L or an applied average ambien emperaure T a,c ( a a hermal conac area c =,.., C or a hermal hea flux J k ( a a hermal conac k =,.., J. The marices M i l (x) represen he model parameers for a chosen se of locaions x, usually he ho spos of he sysem in he chip ceners. The M i l (x) values are obained by linear leas square fis o uni sep responses of FEM-simulaed or meas- Z Y l ( ) / EDA Publishing/THERMINIC 27 -page- ISBN:

2 ured T(x, for individual s l (. Using he expression () for single source uni-sep heaing wih s l ( = Θ( and s m ( = for all m l he emperaure field in case of homogeneous saring emperaure T(x,) = akes he more cusomary appearance of a hermal impedance (heaing curve for uni power): M / zhl ( x, = = R i i i( x) ( e ) (2) wih R i (x) = i M i l (x). zh l (x, denoes he uni sep response of he sysem for heaing only e.g. chip l wih uni srengh of power or one hermal conac area wih a uni sep in ambien emperaure or one hermal conac wih a uni hea flux sep. zh l (x, hus is he generalized hermal impedance for fixed posiion x. I can serve o calculae T(x, for arbirary ime evoluion s l ( by convoluion of s l ( wih he ime derivaive żh l (x, [5, 6], which can also be direcly inferred from (). Having simulaneously several sources s l ( he individual conribuions o he emperaure field are superposed according o (). Eq. (2) is represened by he Foser hermal circui of Fig. 2 wih R i C i = i, hus C i = i /R i. Since () and (2) are valid models for general 3D sysems (wih negligible nonlineariie he circui of Fig. 2 can also represen 3D sysems. Superposiion of several hea sources can be represened by a series connecion of circuis of Fig. 2 [] ogeher wih heir individual hea sources (pseudo-curren source. When wo or more hermal sysems are combined, i is no sraighforward o derive he hermal model of he combined sysem from he known hermal models of he individual sysems. A ypical ask in pracice is o moun a semiconducor module or single chip package on a cooling radiaor and o provide he hermal model for he complee sysem. The model for he combined sysem is no obained by series connecion of he Foser-circui juncion-case for he package and he Foser-circui case-ambien for he radiaor, because he Fosercircui is only a behavioral descripion of he subsysems bu no rue physical descripion. This is easily recognized by he fac ha hea propagaion hrough he Foser circui is insananeous, i.e. he hea flow enering a he lef hand (source side) leaves a he same momen wih equal srengh a he righ hand side by curren coninuiy and would flow ino he series conneced radiaor circui. In realiy i needs some ime for he package o warm up unil hea flow ino he radiaor occurs. The Cauer-circui shown in Fig.3 can accoun for his delay and provides a much more physical descripion of he hea flow pah and i can describe he same hermal impedances Dissipaed power P( T 5 R5 T 4 R4 T 3 R3 T 2 R2 T R C5 C4 C3 C2 C Fig. 2: Foser ype hermal equivalen circui wih applied hea power as curren -source. Budapes, Hungary, 7-9 Sepember 27 Dissipaed power P( T 5 r5 T 4 r4 T 3 r3 T 2 r2 T r c5 c4 c3 c2 c Fig. 3: Cauer ype hermal equivalen circui. When omiing he dashed line, he circui forms a wo-por for connecion o anoher subsysem Cauer-circui. zh( as he Foser-circuis in case of consan righ hand side emperaure. The Cauer-circui is closely relaed o he concep of srucure funcion [7]. The Cauer-circuis being physical models can be series conneced o form Cauer ladders for he oal sysem. When connecing he Cauer-circuis, hey acually form wo-pors conrary o he one-por Foser-circuis [8], Fig.9. However, he mahemaical represenaion of he Cauer form is much more complicaed han he Foser eq. (2) and i is more difficul o deermine he Cauer nework parameers r i, c i by fi. For his reason he hermal impedances in module daa shees are provided by Foser-parameers, which are of lile use for combined sysems. The mehod suggesed in his work proceeds in hree seps: Firs he (case-) inerface beween he wo subsysems (e.g. module and radiaor) is subdivided ino several hermal conac areas, so ha he emperaure a each conac area is approximaely homogeneous. This will be necessary for large modules as in Fig.. The hermal subsysems are described by Foser-models of he form (), (2). In a second sep for each hermal conac area he corresponding Foser-circuis in subsysem and 2 are ransformed in Cauer ladders and conneced a he hermal conacs. The resul is a parallel connecion of Cauer-ladders beween juncion (module chip hea source) and ambien (boom of radiaor or cooling fluid wih consan ambien emperaure). The parallel Cauer ladders are combined in he Laplace-domain by sraighforward algebra o form one hermal impedance, which is ransformed in a hird sep o a Foser-circui juncion-ambien by parial-fraciondecomposiion (secion III). The Foser circuis hus obained characerize he combined sysem hermally by simple mahemaics of he ype of (2). III. TRANSFORMATION OF EQUIVALENT THERMAL CIRCUITS The Foser-Cauer circuis are reaed in works on nework synhesis [8]. In he following a concise derivaion for he Foser - Cauer ransformaion and vice versa will be presened, which leads o fas algorihms so ha a large number of ransformaions can be performed, sufficien for he descripion of whole emperaure fields. The Foser-circui of Fig. 2 can be creaed recursively (or ieraively) by he prescripion expressed in Fig.4. Zh n ( is he one-por impedance of he circui wih n hermal resisors and capaciors R,.., R n, C,.., C n. The iniial impedance Zh ( in Fig. 4 and 5 is zero, i.e. Zh has zero resisance (direc connecion). Generally, he effec of a one-por impedance wih consan reference emperaure T on he righ hand side and emperaure T n ( on he lef hand side where a ime dependen curren -source P n ( is applied, can be represened for he iniial condiion T n ( = ) = T EDA Publishing/THERMINIC 27 -page- ISBN:

3 Budapes, Hungary, 7-9 Sepember 27 by he convoluion inegral: Rn ( T ( τ ) Zhn ( τ (3) Zh n = Cn Zh n ( as used here is he impulse -response (δ-response) and equal o he ime-derivaive of he uni-sep response zh( of (2). The impedance of a simple hermal resisor R wihou capaciance is in his noaion Zh( = δ( R. The circui of Fig. 4 leads o he following equaion sysem for T = and ZhF n ( denoing he Foser-ype impedances: P ( P ( ( T ( T ( )/ R C d n = n = n n n + n ( ( ( ) d ( ( τ ) ZhFn ( τ, ZhFn d ( τ ) ( τ ) τ This deermines ZhF n ( when P n ( and ZhF n- ( are known. The inegro-differenial equaion sysem is ransformed ino an algebraic sysem by applying he Laplace-ransformaion = s L { T( } = T( T( e d wih s = i ω. Under he linear operaion L he convoluion inegrals are ransformed in simple producs P n ( Zh n ( in s- space and ime derivaes ransform o producs wih s: L{dT(/d} = s T(. The Laplace-ransformed equaion sysem for Fig.4 reads: Pn ( = ( ( )/ Rn + Cns ( ( ) ( ( ZhFn (, ( ZhFn P n ( T P n n- T n- Zh n- Fig. 4: Prescripion for recursive generaion of Foser-circui of Fig.2 Eq. (5) leads o a coninued fracion represenaion of ZhC n ( which can also be wrien as raional funcion in s, p n (/q n (, wih polynomial degree of p n ( smaller by one han ha of polynomial q n (. An efficien algorihm for a fas calculaion of he p n (, q n ( can easily be derived. The Cauer-impedance ZhC n ( = p n (/q n ( can be ransformed ino he Foser-form (4) by decomposiion of he raional funcion in parial fracions. A firs he zeroes s k = / k = /(R k C k ) of q n ( = have o be deermined, which can be done symbolically up o n 4 degree and numerically wihou limiaion in n. The coefficiens in (4) can be shown o be /C k = p(s k ) /q (s k ). Thus he Foser R k, C k are obained from he Cauer r k, c k. The symbolic calculaion of he Foser R k, C k (limied o n 4) gives rise o exremely lenghy, unwieldy expressions ha will no be reproduced in his paper, however he numerical evaluaion is done very quickly and precisely. The inverse ransformaion of he Foser-circui ino he Cauer-circui makes use of he recurrence relaion (5) in he form: /ZhC n ( = s c n +/(r n + ZhC n () (6) From his: T n (/P n ( =ZhF n ( = /(s C n + /R n ) + ZhF n- ( and he ZhF n ( can be calculaed very simply recursively o yield he well known form [9]: N ZhFN ( = = /( C = = k k( s sk )), sk / k /( Rk C k) (4) The Cauer-circui of Fig. 3 is generaed by he recursive prescripion of Fig.5, which is expressed analyically: P ( ( T ( T ( )/ r, P ( P ( c d n = n n n n = n + n ( d ( ( τ ) ZhCn( τ, ZhCn d ( τ ) ( τ ) τ and Laplace ransformed reads: Pn ( = ( ( ( )/ rn, Pn ( + cn s ( ( ( ZhCn(, ( ZhCn From his he recursive definiion of he Cauer impedances ZhC n ( resuls: ( / Pn ( = ZhCn( = /( s cn + ) (5) rn + ZhCn Equaing he Foser-impedance (4) wrien as raional funcion ZhF n ( = p n (/q n ( for n =N wih he Cauer ZhC n (, he raional funcion /ZhC n ( = q n (/p n ( is decomposed by he sandard Euklid s algorihm ino a polynomial linear in s and a raional funcion rem n (/p n ( as remainder: /ZhC n ( = q n (/p n ( = s c n + k n + rem n (/p n ( The polynomial degree(rem n ) < degree(p n ). Comparing his expression wih (6), c n has o be idenified wih c n and k n + rem n (/p n ( wih /(r n + ZhC n (). Making use of he ideniy rem ( p ( k n / n n + =, pn( kn pn( + remn( we have: p n ( / (k n p n ( + rem n () = r n + ZhC n (. Zh n = P n ( T P n rn n- T n- cn Zh n- Fig. 5: Prescripion for recursive generaion of Cauer-circui of Fig.3 EDA Publishing/THERMINIC 27 -page- ISBN:

4 Budapes, Hungary, 7-9 Sepember 27 Fig. 6: TO-28AB Package for 2V / 35A Siemens/Infineon single chip IGBT, BUP 37 Using Euklid s decomposiion for he raional funcion on he lef hand side and seing ZhC n ( = p n ( / q n ( he following relaions are obained: r n = /k n, q n ( = k n p n (+ rem n (, p n ( = rem n (/k n. Fig. 7: Thermal impedance juncion-ambien ZhJA for 2V/35A IGBT mouned on radiaor. Dissipaed power 4.73W. Curve fi wih model (2) o daa from measuring signals averaged in small ime inervals. Thus he Cauer r n, c n are deermined for n = N. The new ZhC n ( defined by p n (/q n ( can be used in he same way as ZhF n ( = p n (/q n ( above, in order o deermine he nex Cauer r n, c n. The algorihm coninues unil all Cauer r, c are compued and is erminaed a p ( =. I should be noed ha he Cauer parameers r i, c i are deermined unequivocally by he Foser-Cauer ransformaion indicaing he physical meaning of he Cauer-circui. On he oher hand, he represenaion of he Foser impedances by he R i, C i is ambiguous, since every permuaion of pairs of R i, C i in (2) or Fig.2 leads o a mahemaically idenical expression. In pracice no more han 5 pairs of resisors and capaciors are necessary o represen a hermal impedance. In ess and applicaions one circui represenaion was ransformed ino he oher and hen back again. In every case he original circui parameers were recovered wih perfec accuracy. IV. APPLICATION AND DISCUSSION For packages of small inerface size beween power dissipaing chip and cooling radiaor i is no necessary o subdivide he inerface in several pieces, as suggesed a he end of secion II, because he inerface emperaure is essenially homogeneous. We have performed hermal measuremens wih a TO-28AB package of abou 2 cm 2 inerface area mouned on a closed waer cooled radiaor. The package is shown in Fig. 6 and conains a single Siemens/Infineon 2V/35A IGBT chip BUP 37. The hermal impedance juncion-ambien of he complee se-up, denoed by ZhJA(, is he heaing curve of he chip for applied consan dissipaed chip power of uni srengh (uni sep-response). I is measured by monioring he cooling down of he IGBT from a heaed seady sae afer urn off of he hea generaing IGBT. The emperaure of he IGBT is observed by measuring he chip s on-sae volage for a small impinged consan curren whose hea generaion during cool down is negligible. ZhJA( is obained from he cooling down curve by he superposiion principle of he linear 3Dhea conducion equaion [5]. This is valid under he assumpion ha maerial parameers (hermal conduciviy, specific hea in he se-up and boundary condiions are independen of emperaure, bu also applies for he ypical maerial parameer variaions in a range beween 3 K and 4 K. Fig. 7 shows he daa poins for ZhJA( inferred from he measured cooling down curve. The daa poins are arihmeic averages of he original digialized measuremen signals in small ime inervals (one poin for each inerval). The curve fied o he daa was obained wih a nonlinear fi-rouine working according o he Levenberg-Marquard mehod [2] o fi he model (2) wih 9 pairs R i, C i (C i = i /R i ). Nearly he same fiing curve was obained when fiing he original daa wih over 2, daa poins, wih only small deviaions a small imes (below m and emperaures where he measuremen is no as accurae (Fig. 8). The hea conducion equaion leads in agreemen wih (3) o he convoluion inegral represenaion of he chip (juncion) emperaure for arbirary chip-power dissipaion profile P(: TJ ( Ta = P( τ ) ZhJA & ( τ (7) (T a = ambien emperaure; ŻhJA ime derivaive of ZhJA). Because of he direc derivaion of (7) from he hea conducion equaion, is validiy is resriced o a cerain class of boundary condiions and o homogeneous saring condiions Fig 8: Same as in Fig. 7, bu curve fi o original daa wih over 2, poins. EDA Publishing/THERMINIC 27 -page- ISBN:

5 T(x, ) = T a as deailed in [5]. Also, (7) holds only, if he power densiy disribuion in he chip is allowed o change is overall srengh wih ime bu no is spaial disribuion. Manufacurers daa shees for power devices usually conain only he hermal impedance of he package or module juncion o case ZhJC( = (T J ( T Junc T c ()/P, where he case emperaure T c ( depends on he seleced locaion, e.g. in he middle of he inerface beween module-base-plae and cooling radiaor. The hermal resisance of he inerface (ypically made up of a hermal grease) has o be included in he hermal impedance of he radiaor. Someimes in he lieraure T c ( is measured simulaneously wih T J ( o obain heaing curves wih consan heaing. The ZhJC( inferred from he difference T J ( T c ( is he difference of wo monoonously increasing funcions. However, his difference and he ZhJC( hus defined are no necessarily monoonously increasing. Examples for nonmonoonous ZhJC( in form of an anomalous bump have been observed in he lieraure [sof]. A meaningful definiion of ZhJC( for hermal characerisaion of he package, which is independen of he choice of he cooling radiaor or cooling condiions, can only be obained for consan case emperaure T c ( = cons. On he oher hand, his condiion is difficul o realise experimenally, because an ideal cooler would be required o keep he inerface emperaure consan in ime. This could be achieved by an acive hermoelecric cooler. Our mehod proceeds in he following way and infers he unknown ZhJC( for consan case emperaure from he measured ZhJA(, when only he seady sae case emperaure (or he hermal resisance of he radiaor) is known: The case emperaure T c is measured in seady sae before he cool down sars by a hermocouple pressed a he package boom side hrough a hole in he radiaor. The measured ZhJA( is represened by he Foser-model (2) as described (Fig. 7). Then he Foser R i, C i are ransformed in Cauer r i, c i wih he algorihm of secion III. The resuling Cauer ladder juncionambien is cu ino wo pieces juncion-case and case-ambien along he line shown in Fig. 9, where he sum of he resisors Budapes, Hungary, 7-9 Sepember 27 module T Case radiaor TAmbien Fig. 9: Decomposiion of Cauer-circui for ZhJA ino ZhJC and ZhCA along dashed line, where sum of resisors from ambien side = Rh-radiaor. Also prescripion for combining ZhJC and ZhCA o ZhJA. couned from he righ hand (ambien side becomes larger han Rh radiaor = (T c T a ) / P. The lef par of he divided Cauer circui is he ZhJC(, he Cauer circui righ from he cuing line forms he hermal impedance of he radiaor ZhCA(. I should be noed ha he impedance for ZhCA( consiss of a Cauer ladder as in Fig. 3 ogeher wih he corresponding par of he hermal resisor divided by he cuing line aached o he lef hand node. Due o his hermal series resisor he impedance ZhCA( performs a sep a = and rises a he beginning wih infinie seepness conrary o ZhJC( or ZhJA( which rises linearly for < -4 sec in case of a volume hea source in he chip []. The hea source applied o he cooling radiaor is no volume hea source in he radiaor bu a hea flux a is surface. In his case he emperaure response o a uni sep in he hea flux a he radiaor surface behaves as [22, ]. Therefore he hermal series resisor in he model for ZhCA( describes he physics more correcly, han he pure Cauer-circui of Fig. 3. The ransformed Foser- ZhCA has he same hermal series resisor as he Cauercircui. The Cauer circuis hus obained for ZhJC( and ZhCA( were ransformed ino Foser-circuis by use of he algorihm of secion III. The Foser circui elemens - and also he Cauer elemens - of ZhJC( provide a suiable represenaion for he hermal characerisaion in manufacurers daa shees. In order o obain from he Foser values he complee impedance ZhJA( of a combined sysem, he subsysem Foser-circuis have o be ransformed in Cauer-circuis which are conneced according o Fig. 9 a he inerface node. Afer his he resuling Cauer-ladder juncion-ambien is ransformed in Foserform for ease of mahemaical presenaion. Performing hese seps we obained he original Foser elemens for ZhJA( (Fig. 7) wih perfec accuracy. Fig. : ZhJC resuling from ZhJA (dashed line) of Fig. 7 according o he prescripion of Fig. 9. Fig. : Direc addiion of he ZhJC, ZhCA consruced according o Fig. 9, leads o wrong predicion for ZhJA (dashed line). EDA Publishing/THERMINIC 27 -page- ISBN:

6 The resul of he consrucion of ZhJC( from ZhJA( by his prescripion is displayed in Fig. ogeher wih he original ZhJA(. The difference ZhJA( - ZhJC( for large imes is equal o he saic hermal resisance of he radiaor Rh radiaor. The simple addiion of he radiaor impedance ZhCA( and of ZhJC( - which corresponds o a series connecion of he Foser-circuis of he respecive impedances - in order o obain ZhJA( gives a grossly wrong resul, as can be seen from Fig.. I is indispensable, firs o ransform he ZhJC, ZhCA in Cauer-ladders and o combine he Cauer-ladders o obain he correc ZhJA. REFERENCES [] C.J.M. Lasance, D. den Herog, and P. Sehouwer, Creaion and Evaluaion of Compac Models for Thermal Characerizaion Using Dedicaed Opimizaion Sofware, in Proc. IEEE SEMI-THERM XV, San Diego, USA, 999, pp [2] H. Rosen, C.J.M. Lasance, and J. Parry, The world of hermal characerizaion according o DELPHI- Par I: Background o DELPHI and Par II: Experimenal and Numerical Mehods, IEEE Trans. Comp., Hybrids, Manufac. Technol., vol. 2, pp , Dec [3] M.N. Sabry, Saic and dynamic hermal modeling of ICs, Microelecronic J. vol. 3, pp.85-9, 999. [4] H. Pape and G. Noebauer, Generaion and verificaion of boundary independen compac hermal models for acive componens according o he DELPHI / SEED mehods, in Proc. IEEE SEMI-THERM XV, San Diego, CA, 999, pp [5] Y.C. Gersenmaier, H. Pape, and G. Wachuka, Rigorous model and nework for saic hermal problems, Microelecronic J., vol. 33, pp.7-78, 22. [6] F. Chrisiaens, B. Vandevelde, E. Beyne, R. Merens, and J. Berghmans, A Generic Mehodology for Deriving Compac Dynamic Thermal Models, Applied o he PSGA package, IEEE Trans. on Componens, Packaging and Manufacuring Technology, Par A Vol 2, No.4, pp , 998. [7] M. Rencz and V. Székely, Dynamic hermal mulipor modeling of IC packages, IEEE Trans. on Comp. Packag. Technol., vol 24, No.4, pp , 2. [8] Y.C. Gersenmaier and G. Wachuka, Rigorous model and nework for ransien hermal problems, Microelecronic J., vol. 33, pp , 22. Budapes, Hungary, 7-9 Sepember 27 [9] D. Schweizer and H. Pape, Boundary Condiion Independ-en Dynamic Thermal Compac Models of IC-Packages, Proc. 9h THERMINIC, Aix-en-Provence, France, Sep. 23, pp [] W. Bay, C. Chrisofferson, A.J. Panks, S. David, C.M. Snowden, and M.B. Seer, Elecrohermal CAD of Power De-vices and Circuis wih Fully Physical Time-Dependen Compac Thermal Modeling of Complex Nonlinear 3-D Sysems, IEEE Trans. Comp. Packag. Technol., Vol.24, No.4, pp , 2. [] Y.C. Gersenmaier, G. Wachuka, Efficien Calculaion of Transien Temperaure Fields Responding o Fas Changing Hea-sources Over Long Duraion in Power Elecronic Sysems, IEEE Trans. Comp. Packag. Technol., vol.27, pp.4-, Mar. 24. [2] Y.C. Gersenmaier, A. Casellazzi, G. Wachuka, Elecro-hermal Simulaion of Mulichip-Modules wih Novel Transien Thermal Model and Time-Dependen Boundary Condiions, IEEE Trans. Power Elecronics, vol.2, pp.45-55, Jan. 26. [3] L. Codecasa, D. D Amore, P. Maffezzoni, Nonlinear Pro-jecion-Based Approach for Generaing Compac Models of Nonlinear Thermal Neworks, Proc. 2h THERMINIC, Nice, 26, pp [4] M. Rencz, V. Székely, Sudies on he Nonlineariy Effecs in Dynamic Compac Model Generaion of Packages, IEEE Trans. Comp. Packag. Technol., vol. 27, pp. 24-3, Mar. 24. [5] Y.C. Gersenmaier and G.Wachuka, A new Procedure for he Calculaion of he Temperaure Developmen in Elecronic Sysems, in Proc. EPE'99 Conf., Lausanne, Swizerland, 999. [6] Y.C. Gersenmaier and G. Wachuka, Calculaion of he emperaure developmen in elecronic sysems by convoluion inegrals, Proc. IEEE SEMI-THERM XVI, San Jose, USA, 2, pp [7] V. Székely, T. V. Bien, Fine Srucure of Hea Flow Pah in Semiconducor Devices, Solid-Sae Elecronics, vol. 3, no. 9, pp , 988. [8] L. Weinberg, Nework Analysis and Synhesis, New York, McGraw-Hill, 962. [9] V. Székely, Idenificaion of RC neworks by deconvoluion: chances and limis, IEEE Trans. Circuis Sys.-I, vol. CAS-45, pp , March 998. [2] S. Wolfram, Mahemaica, Wolfram Research, Champaign, 999 [2] J.W. Sofia, IEEE Trans. Comp., Pack., Manufac. Tech.-Par A, vol. 8, no., pp.39-47, 995. [22] H.S. Carslaw, J.C. Jaeger, Conducion of Hea in Solids, 2nd ediion, Oxford Universiy Press, 959. EDA Publishing/THERMINIC 27 -page- ISBN: