Thermal Spreading Resistance of Eccentric Heat Sources on Rectangular Flux Channels

Transcription

1 Y. S. Muzychka Assstant Professor, Faculty of Engneerng and Appled Scence, Memoral Unversty of ewfoundland, St. John s, ewfoundland, Canada A1B 3X5 J. R. Culham Assocate Professor, Mem. ASME M. M. Yovanovch Dstngushed Professor Emertus, Fellow ASME Mcroelectroncs Heat Transfer Laboratory, Department of Mechancal Engneerng, Unversty of Waterloo, Waterloo, Ontaro, Canada 2L 3G1 Thermal Spreadng Resstance of Eccentrc Heat Sources on Rectangular Flux Channels A general soluton, based on the separaton of varables method for thermal spreadng resstances of eccentrc heat sources on a rectangular flux channel s presented. Solutons are obtaned for both sotropc and compound flux channels. The general soluton can also be used to model any number of dscrete heat sources on a compound or sotropc flux channel usng superposton. Several specal cases nvolvng sngle and multple heat sources are presented. DOI: / Keywords: Resstance Conducton, Electroncs Coolng, Heat Spreaders, Heat Snks, Spreadng Introducton Thermal spreadng resstance results from dscrete heat sources n many engneerng systems. Typcal applcatons nclude coolng of electronc devces both at the package and system level, and coolng of power sem-conductors usng heat snks. In these applcatons, heat dsspated by electronc devces s conducted through electronc packages nto prnted crcut boards or heat snk baseplates whch are convectvely cooled. In heat snk applcatons the convectve flm resstance s replaced by an effectve extended surface flm coeffcent. Analytcal results for an eccentrc heat source on a fnte rectangular flux channel are obtaned for sotropc and compound systems. These solutons may be used to model sngle or mult-source systems by means of superposton. The present approach dffers from other methods 1 3, n that the heat source specfcaton s ncorporated n the defnton of the thermal boundary condtons rather than n the governng partal dfferental equaton. Ths results n analytcal expressons whch can be easly manpulated n most advanced mathematcal software packages 4 7. The general soluton for the spreadng resstance of a sngle constant flux eccentrc heat source wth convectve or conductve coolng at one boundary wll be presented, see Fg. 1. A revew of the lterature reveals that ths partcular confguraton has not yet been analyzed 8. The two-dmensonal eccentrc strp heat source for a sem-nfnte flux channel was obtaned by Vezroglu and Chandra 9. Whle the soluton for a centrally located heat source on an sotropc rectangular plate was obtaned by Krane 10. Recently, Yovanovch et al. 11 obtaned a soluton for a centrally located heat source on a compound rectangular flux channel. The general soluton wll depend on several dmensonless geometrc and thermal parameters. In general, the total resstance s gven by R T R 1D R s (1) R s s the spreadng resstance and R 1D s the one dmensonal resstance of the system gven by R 1D t 1 k 1 A b 1 ha b (2) Contrbuted by the Electronc and Photonc Packagng Dvson for publcaton n the JOURAL OF ELECTROIC PACKAGIG. Manuscrpt receved by the EPPD Dvson February 26, Guest Edtors: Y. Muzychka and R. Culham. A b ab, and h s a heat transfer coeffcent whch may be a contact conductance or effectve fn conductance. The total thermal resstance s defned as R T T st f T s s the average source temperature gven by T s 1 A s A Tx,y,0dA s (4) s The thermal resstance for the confguraton shown n Fg. 1 s a functon of R f a,b,c,d,x c,y c,t 1,h,k 1 (5) Problem Statement The governng equaton for the system shown n Fg. 1 s Laplace s equaton 2 T 2 T x 2 2 T y 2 2 T z 2 0 (6) whch s subjected to a unform flux dstrbuton T /A s (7) k zz0 1 wthn the heat source area, A s cd, and T zz0 (3) 0 (8) outsde the heat source area, and a convectve or mxed boundary condton on the bottom surface T zzt 1 h k 1 Tx,y,t 1 T f (9) In extended surface applcatons such as heat snks, the value of h s replaced by an effectve value whch accounts for both the heat transfer coeffcent on the fn surface and the ncreased surface area h eff 1 (10) A b R fns 178 Õ Vol. 125, JUE 2003 Copyrght 2003 by ASME Transactons of the ASME

2 and T 1 x,y,t 1 T 2 x,y,t 1 (13) k 1 T 1 z zt 1 k 2 T 2 z zt 1 (14) whle along the bottom surface zt 1 t 2, the boundary condton to be satsfed becomes T 2 z zt 1 t 2 h k 2 T 2 x,y,t 1 t 2 T f (15) The full soluton s obtaned for the sotropc case, however, t may easly be appled to the compound system wth only slght modfcaton, usng the results of Yovanovch et al. 11. General Soluton The soluton for the sotropc plate may be obtaned by means of separaton of varables The soluton s assumed to have the form (x,y,z)x(x)*y(y)*z(z), (x,y,z) T(x,y,z)T f. Applyng the method of separaton of varables yelds the followng general soluton for the temperature excess n the plate whch satsfy the thermal boundary condtons along (x 0, xa) and (y0, yb) x,y,za 0 B 0 z m1 cosxa 1 coshzb 1 snhz n1 cosya 2 coshzb 2 snhz Fg. 1 Isotropc plate wth eccentrc heat source m1 n1 cosxcosya 3 coshz Along the edges of the plate, the followng condtons are also requred: and T xx0,a T yy0,b 0 (11) 0 (12) The general soluton for the total thermal resstance and temperature dstrbuton wll be obtaned for the system shown n Fg. 1. In a later secton, the soluton wll be extended to compound systems as shown n Fg. 2. In a compound system Laplace s equaton must be solved n each layer. In addton to the boundary condtons specfed for the sotropc system, the followng condtons wth perfect contact at the nterface are requred: Fg. 2 Compound plate wth eccentrc heat source B 3 snhz (16) m/a, n/b, and 2 2. The soluton contans four components, a unform flow soluton and three spreadng or constrcton solutons whch vansh when the heat flux s dstrbuted unformly over the entre surface z 0. Snce the soluton s a lnear superposton of each component, they may be dealt wth separately. Applcaton of the boundary condtons n the z drecton wll yeld solutons for the unknown constants. Applcaton of the thermal boundary condton at zt 1 for an sotropc rectangular plate yelds the followng result for the Fourer coeffcents: B A 1,2,3 (17) snht 1h/k 1 cosht 1 (18) cosht 1 h/k 1 snht 1 and s replaced by,, or, accordngly. The fnal Fourer coeffcents A m, A n, and A mn are obtaned by takng Fourer seres expansons of the boundary condton at the surface z0. Ths results n or and A m A 1 bck 1 m m 2 sn 2X cc 2 X c c/2 Xc cos c/2 m xdx 0 a cos 2 m xdx m sn 2X cc 2 m abck 1 m 2 m (19) (20) Journal of Electronc Packagng JUE 2003, Vol. 125 Õ 179

3 or and A n A 2 adk 1 n n 2 sn 2Y cd 2 Y c d/2 Y cos c d/2 n ydy 0 b cos 2 n ydy n sn 2Y cd 2 n abdk 1 n 2 n (21) (22) Y c d/2 X Y c c/2 A 3 cdk 1 m,n m,n cd/2 Xc cos c/2 m xcos n ydxdy b 0 a 0 cos 2 m xcos 2 n ydxdy (23) or 16 cos m X c sn 1 2 mc cos n Y c sn 1 2 nd A mn abcdk 1 m,n m n m,n (24) Fnally, values for the coeffcents n the unform flow soluton are gven by A 0 ab t 1 k 1 1 h (25) and B 0 (26) k 1 ab Mean Temperature Excess. The general soluton for the mean temperature excess of a sngle heat source may be obtaned by ntegratng Eq. 16 over the heat source area. Carryng out the necessary ntegratons leads to the followng expresson for the mean source temperature: 1D 2 m1 2 n1 4 m1 n1 A m cos m X c sn 1 2 mc m c A n cos n Y c sn 1 2 nd n d A mn cos n Y c sn 1 2 nd cos m X c sn 1 2 mc m c n d (27) 1D ab t 1 k 1 1 h (28) for an sotropc system. Thermal Spreadng Resstance. The thermal spreadng resstance may now be computed usng the mean temperature excess. In general, the total resstance as defned by Eq. 3, results n R T R 1DR s (29) R 1D s the one-dmensonal thermal resstance and R s s the thermal spreadng resstance. The thermal spreadng resstance component s defned by the three seres soluton terms n Eq. 27. Fg. 3 Compound Systems. In many applcatons an nterface materal may be added to reduce thermal contact conductance and/or promote thermal spreadng. The soluton obtaned for the sotropc rectangular flux channel may be used for a compound flux channel wth only mnor modfcatons. In Yovanovch et al. 11 the authors obtaned a soluton for a compound rectangular flux channel. It s not dffcult to show that n Eqs. 20, 22, 24 can be replaced by e4t 1e 2t 1ϱe 22t 1 t 2 e 2t 1 t 2 e 4t 1e 2t 1ϱe 22t 1 t 2 e 2t 1 t 2 ϱ h/k 2 and 1 h/k 2 1 (30) wth k 2 /k 1, and s replaced by,, or, accordngly. Ths smple extenson s possble, snce the effect of the addtonal layer results from solvng for the unknown coeffcents by applcaton of the boundary condtons n the z drecton. The general soluton for all but one of the Fourer coeffcents s dentcal to the case for a central source. Snce the spreadng resstance s based upon the mean source temperature at the surface of the flux channel, t s not necessary to resolve the complete system of equatons. However, complete soluton for all coeffcents s requred for calculatng the temperature wthn the sold. Addtonally, 1D ab t 1 k 1 t 2 k 2 1 h (31) for a compound system. Applcaton of the above results s only vald for computng the temperature dstrbuton at the surface of the baseplate and to compute the spreadng resstance. Equaton 30 s merely the recprocal of a smlar expresson reported n Yovanovch et al. 11. Specal Cases Specal cases of a plate wth eccentrc heat source Several specal cases of an eccentrc heat source may be obtaned from the general soluton. These are shown n Fg. 3. Soluton for a central heat source on an sotropc plate was obtaned by Krane 10. However, the results are only presented for the thermal resstance based upon the maxmum or centrodal temperature dfference. Recently, Yovanovch et al. 11 obtaned the soluton for the spreadng resstance of a centrally located heat source on a compound plate. A specal case of the soluton of Yovanovch et al. 11 s that for an sotropc plate; see Fg Õ Vol. 125, JUE 2003 Transactons of the ASME

4 Fg. 5 Compound plate wth central heat source Addtonally, the remanng cases n Fg. 3 may also be obtaned from the soluton for a central heat source usng the method of mages. Central Heat Source. The spreadng resstance of Yovanovch et al. 11 s obtaned from the followng general expresson whch shows the explct relatonshps wth the geometrc and thermal parameters of the system accordng to the notaton n Fgs. 4 and 5: R s 1 2a 2 cdk 1 m1 Fg. 4 Isotropc plate wth central heat source 1 n a 2 b 2 cdk 1 m1 sn 2 a m 1 3 m m 2b 2 cdk 1 n1 n1 sn 2 b n n 3 sn 2 a m sn 2 b n 2 m 2 n m,n m,n (32) e2t 11e 2t h/k 1 e 2t 11e 2t (33) h/k 1 If the system s composed of two layers, then e4t 1e 2t 1ϱe 22t 1 t 2 e 2t 1 t 2 e 4t 1e 2t 1ϱe 22t 1 t 2 e 2t 1 t 2 (34) ϱ h/k 2 and 1 h/k 2 1 wth k 2 /k 1. The egenvalues for these solutons are: m m/c, n n/d and m,n 2 m 2 n. Edge and Corner Heat Sources. The soluton obtaned by Yovanovch et al. 11 may also be used to model the three addtonal specal cases shown n Fg. 3. By means of symmetry, the soluton for the spreadng resstance may be obtaned by consderng that each of the specal cases represents an element of the system wth a centrally located heat source. For an edge source, the resstance s gven by R s 2R, and for a corner heat source the total resstance s gven by R s 4R, R s the resstance of the system composed by mrrorng the mages of the edge or corner heat sources to obtan a system wth a central heat source. Sem-Infnte Isotropc Flux Channel. Addtonal results may be obtaned for sem-nfnte flux channels for the case t 1 and the effect of the conductance h s no longer a factor. The soluton for a sem-nfnte flux channel s obtaned when the parameter 1 (35) Sem-Infnte Compound Flux Channel. The general expresson for reduces to a smpler expresson when t 2, see Fg. 2. The soluton for ths partcular case arses from Eq. 30 wth e2t 11e 2t 11 e 2t 11e 2t (36) 11 the nfluence of the convectve conductance has vanshed, but the nfluence of the substrate remans. Eccentrc Strp Solutons. Fnally, solutons for both sotropc and compound eccentrc strps may be obtaned from the general soluton. If the dmensons of the heat source extend to two of the boundares to form a contnuous strp, the general soluton smplfes consderably. In Eq. 27, the general soluton conssts of four terms, a unform flow component, two strp solutons sngle summatons and a rectangular source soluton double summaton. For the case of an eccentrc strp, one need only Journal of Electronc Packagng JUE 2003, Vol. 125 Õ 181

5 Table 1 Results for case 1 a Tˆ 1 T 1 Tˆ 2 T 2 T b Present Culham whch may be wrtten 1 j 1 A x,y,0da j 1 (40) j A j Usng Eqs. 27 and 40 results n the followng expresson for the mean temperature of the jth heat source: T jt f 1 (41) Fg. 6 consder the approprate strp soluton and the unform flow soluton. The remanng summatons fall out of the soluton when b d or ca. In the case of sem-nfnte eccentrc strp solutons, Eqs. 35 and 36 also hold. Multple Heat Sources If more than one heat source s present see Fg. 6, the soluton for the temperature dstrbuton on the surface of the crcut board or heat snk may be obtaned usng superposton. For dscrete heat sources, the surface temperature dstrbuton s gven by Tx,y,0T f 1 x,y,0 (37) s the temperature excess for each heat source by tself. The temperature excess of each heat source may be computed usng Eq. 16 evaluated at the surface x,y,0a 0 m1 A m m1 n1 cosx n1 A n cosy A mn cosxcosy (38) wth defned by Eq. 18 or 30 and A 0 1D gven by Eq. 28 or 31. The mean temperature of an arbtrary rectangular patch of dmensons c j and d j, located at X c, j and Y c, j may be computed by ntegratng Eq. 37 over the regon A j c j d j j 1 A j A Plate wth multple heat sources da j 1 j A j A j 1 x,y,0da j (39) A o 2 m1 2 n1 A n A m cos m X c, j sn 1 2 mc j m c j cos n Y c, j sn 1 2 nd j n d j 4 m1 n1 cos n Y c, j sn 1 2 nd j cos m X c, j sn 1 2 mc j m c j n d j A mn Equaton 41 represents the sum of the effects of all sources over an arbtrary locaton. Equaton 41 s evaluated over the regon of nterest c j, d j located at X c, j, Y c, j, wth the coeffcents A 0, A m, A n and A mn evaluated at each of the th source parameters. The coordnate system s based on the orgn placed at the lower left corner of the plate, and each source s located usng the coordnates of the centrod. Applcaton of Results To demonstrate the usefulness of the present approach, several examples of systems contanng multple sources are presented. Frst, an example s gven whch shows the effect of a heat spreader on a low conductvty substrate. ext, the method s appled to model a heat snk contanng a number of dscrete heat sources unformly located on the baseplate. Fnally, a unform flux heat source of complex shape s analyzed. Case 1. In the frst case, the dmensons of a plate or crcut board are: a300 mm, b300 mm, t 1 10 mm, h10 W/m 2 K and k10 W/mK, wth T f 25 C. Two heat sources havng dmensons c25 mm and d25 mm each. The frst source wth a power of 10 W s located at X c Y c 90 mm and the second havng a power of 15 W at X c Y c 210 mm. The basc equatons may be programmed nto any symbolc or numercal mathematcs software package. For the present calculatons, the symbolc mathematcs program Maple V 4 was employed. A total of 50 terms were used n each of the sngle and double summatons. The results for the mean T and centrodal Tˆ temperatures of the frst case are presented n Table 1. In ths example the centrodal temperatures of each heat source were found to be C and C. A three-dmensonal plot of the surface temperature profle s gven n Fg. 7. Comparsons wth a generalzed Fourer seres approach of Culham and Yovanovch 16 yelds C and C. The prmary dfference between the present ap- 182 Õ Vol. 125, JUE 2003 Transactons of the ASME

6 Table 2 Results for case 1 b Tˆ 1 T 1 Tˆ 2 T 2 T b Present Culham proach and that of Culham and Yovanovch 16 s that the present work yelds smplfed expressons whch may be easly programmed n any Mathematcs or Spreadsheet software, as the latter uses a numercal least squares approxmaton to solve for a mxed boundary value problem. For the same confguraton, a thn, t2 mm, hghly conductve layer k350 W/mK, s added to the orgnal substrate and the problem reanalyzed. The results are summarzed n Table 2. In ths example the centrodal temperatures of each heat source were found to be and C. Comparsons wth a generalzed Fourer seres approach of Culham and Yovanovch 16 yelds and C. A three-dmensonal plot of the surface temperature profle s gven n Fg. 8. It s clearly seen that addng a spreader has reduced the maxmum source temperatures consderably and equalzed the temperature. Case 2. In ths example, the baseplate of a heat snk s to be analyzed. The dmensons of the baseplate are a50 mm, b 50 mm, k150 W/mK, t10 mm, and an effectve extended surface heat transfer coeffcent h400 W/m 2 K. Four sources havng the characterstcs summarzed n Table 3, are attached to the baseplate assumng neglgble contact or nterface resstance. The temperature results are summarzed n Table 4 and n Fg. 9. Case 3. In the fnal example, a heat source wth a complex shape s analyzed by the present approach. The source s composed of fve square heat sources each havng dmensons c 20 mm by d20 mm and dsspatng 5 W. The heat sources are arranged n the form of a cross n the center of a plate havng dmensons a100 mm by b100 mm, thckness of t10 mm. Ths results n an rregular shaped soflux heat source whch cannot be solved usng conventonal approaches. The thermal conductvty of the plate s k50 W/mK, whle h50 W/m 2 K, and T f 25 C. The value of the maxmum temperature at the center of Fg. 7 Surface temperature for an sotropc plate wth two heat sources Fg. 8 Surface temperature for a compound plate wth two heat sources Journal of Electronc Packagng JUE 2003, Vol. 125 Õ 183

7 Table 3 Source characterstcs for case 2 W c mm d mm X c mm Yc mm Source Source Source Source Table 4 Source temperatures for case 2 Source Source Source Source Tˆ T the plate s found to be C. The area weghted mean source temperature s found to be C. The thermal contour plot s gven n Fg. 10. Dscusson The general soluton obtaned may easly be coded n a number of ways. The smplest approach s through the use of mathematcal programs 4 7. These packages allow for symbolc and numercal computaton of mathematcal expressons. They also provde graphcal functons for generatng three-dmensonal plots such as those presented earler. One advantage of these packages s the mnmal effort requred to enter the basc equatons. Computaton tme vares among packages and s also dependent upon the number of sources specfed. The present results were obtaned usng Maple V6 4 wth 50 terms n each of the summatons. A sngle source calculaton typcally requred a few seconds. Reasonable convergence wth 50 terms s obtaned for most problems. Another method of computaton whch was assessed s the use of computer languages such as C or Basc. Computaton tme s much faster wth a compled code, however, a consderable amount of code s requred to acheve the same results as those produced usng mathematcal software. The method s also amenable to spreadsheet calculatons wth or wthout the use of macros. However, the use of macros allows for easer development. In addton to provdng detals of the surface temperature dstrbuton and centrodal and mean source temperatures, the effectve thermal resstance may also be computed for each source. Ths approach was not taken n the present work, snce no unque value of thermal resstance s defnable when more than one source s present. The locaton and strength of each addtonal heat source wll affect the value of thermal resstance for a partcular source of nterest. Fnally, f the ambent temperature ncreases as a result of heat transfer due to the flm coeffcent h, a wake effect may be approxmated n the fnal soluton by defnng an ambent temperature whch s a functon of flow poston T f T x (42) ṁc p L T s the nlet temperature, ṁ s the mass flow through the system, s the sum of all heat sources, and x/l the local poston n the flow drecton. Summary and Conclusons General expressons for determnng the spreadng resstance of an eccentrc soflux rectangular source on the surface of fnte sotropc and compound rectangular flux channels were presented. The soluton for the temperature at the surface of a rectangular flux channel was also presented. It was shown that ths soluton may be used to predct the centrodal temperatures for any number of heat sources usng superposton. Addtonal specal cases of spreadng resstance from sngle eccentrc heat sources on sotropc, compound, fnte, and sem-nfnte flux channels were also presented. Fnally, t was shown that the soluton for a central heat source may be used to compute the spreadng resstance for corner and edge heat sources usng the method of mages. Several applcatons of the general soluton to systems wth multple heat sources were also gven. Fg. 9 Surface temperature for a heat snk wth four heat sources 184 Õ Vol. 125, JUE 2003 Transactons of the ASME

8 Fg. 10 Surface temperature for a complex heat source Acknowledgments The authors acknowledge the fnancal support of the atural Scences and Engneerng Research Councl of Canada. The authors would also lke to thank Dave Fast of PowerSnk Technologes Inc., for demonstratng that the present approach can be coded n a Mcrosoft Excel spreadsheet usng Vsual Basc Macros. omenclature a, b, c, d lnear dmensons, m A b baseplate area, m 2 A s heat source area, m 2 A 0, A m, A n, A mn Fourer coeffcents B Bot no., hl/k C p heat capacty, J/Kg K h contact conductance or flm coeffcent, W/m 2 K ndex denotng layers 1 and 2 k, k 1, k 2 thermal conductvtes, W/m K L arbtrary length scale, m ṁ mass flow rate, kg/s m, n ndces for summatons no. of heat sources heat flow rate, W q heat flux, W/m 2 R thermal resstance, K/W R* dmensonless thermal resstance, krl R 1D one-dmensonal resstance, K/W R s spreadng resstance, K/W R T total resstance, K/W t, t 1, t 2 total and layer thcknesses, m T 1, T 2 layer temperatures, K T s mean source temperature, K T f snk temperature, K T nlet or ntal temperature, K Xc, Yc heat source centrod, m m,n egenvalues, 2 2 m n n egenvalues, (n/b) relatve contact sze, b/a References temperature excess, TT f,k mean temperature excess, T T f,k ˆ centrodal temperature excess, Tˆ T f,k b mean base temperature excess, T bt f, K relatve conductvty, k 2 /k 1 m egenvalues, (m/a) spreadng functon relatve thckness, t/l dummy varable, m 1 1 Ellson, G., 1984, Thermal Computatons for Electronc Equpment, Kreger Publshng, Malabar, FL. 2 Kokkas, A., 1974, Thermal Analyss of Multple-Layer Structures, IEEE Trans. Electron Devces, Ed-2114, pp Hen, V. L., and Lenz, V. D., Thermal Analyss of Substrates and Integrated Crcuts, pp , Bell Telephone Laboratores, unpublshed report. 4 Maple V Release 6, 2000, Waterloo Maple Software, Waterloo, O, Canada. 5 Mathcad Release 6, 1995, Mathsoft Inc., Cambrdge, MA. 6 Mathematca, 1996, Release 3, Wolfram Research, Champagn-Urbana, IL. 7 MatLab Release 4, 1997, Mathworks Inc., atck, MA. 8 Yovanovch, M. M., 1998, Chapter 3: Conducton and Thermal Contact Resstance Conductance, Handbook of Heat Transfer, eds., W. M. Rohsenow, J. P. Hartnett, and Y. L. Cho, McGraw-Hll, ew York, Y. 9 Verzroglu, T.., and Chandra, S., 1969, Thermal Conductance of Two Dmensonal Constrctons, Prog. Astronaut. Aeronaut., Vol. 21 Thermal Desgn Prncples of Spacecraft and Entry Bodes, ed., G. T. Bevns, pp Krane, M. J. H., 1991, Constrcton Resstance n Rectangular Bodes, ASME J. Electron. Packag., 113, pp Yovanovch, M. M., Muzychka, Y. S., and Culham, J. R., 1999, Spreadng Resstance of Isoflux Rectangles and Strps on Compound Flux Channels, J. Thermophys. Heat Transfer, 13, pp Arpac, V., 1966, Conducton Heat Transfer, Addson-Wesley, ew York, Y. 13 Carslaw, H. S., and Jaeger, J. C., 1959, Conducton of Heat n Solds, Oxford Unversty Press, Oxford, UK. 14 Ozsk,. A., 1980, Heat Conducton, John Wley and Sons, Inc., ew York, Y. 15 Lukov, A. V., 1968, Analytcal Heat Dffuson Theory, Academc Press, ew York, Y. 16 Culham, J. R., and Yovanovch, M. M., 1997, Thermal Characterzaton of Electronc Packages Usng a Three-Dmensonal Fourer Seres Soluton, The Pacfc Rm/ASME Internatonal, Intersocety Electronc and Photonc Packagng Conference, ITERpack 97, Kohala Coast, Island of Hawa, June Journal of Electronc Packagng JUE 2003, Vol. 125 Õ 185