Transmission Line Matrix: A Tool for Modeling Thermo-Electric properties of Materials and Devices

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1 Jounal of New Technology an Maeals JNTM ol. 02, N 01 (2012)07-12 OEB Unv. Publsh. Co. Tansmsson Lne Max: A Tool fo Moelng Themo-Elecc popees of Maeals an Devces A. Saane*, S. Mmoun an. Houcne CaSCCE Laboao ENSET-Oan, B.P 1523 M Naoue-Oan, Algea * Emal: saaneae@yahoo.com eceve: Dec 29, 2011; accepe as plenay al of NMCA2011 Absac Tansmsson-Lne-Max (TLM) meho, ognally evelope n 1971 as a numecal echnque fo moelng elecomagnec wave popagaon, has snce been esablshe as a poweful echnque o suy ffuson poblems, vbaon, hea ansfe, elecomagnec compabl aa, ec. The TLM meho s a me an space scee meho ha solves fel poblems usng he ccu equvalen. I assembles a lace of scee pons n space as one-mensonal lnes an efnes he ansmsson max beween lace pons, so ha successve calculaons can be pefome. The physcal vaable s moele as a sequence of volage pulses avellng hough hs newo of ansmsson lnes. The TLM oune opeaes on he avellng, scaeng, an connecng of hese pulses n he newo. The ansmsson lnes n he moel ac as elay lnes, wh he noe mpulse populaon beng he scee soluon a each me sep. TLM s a scee moel whch can be solve exacly snce appoxmaons ae only nouce a he scesaon sage. Ths s o be conase wh he aonal appoach n whch an ealze connuous moel s fs obane an hen hs moel s solve appoxmaely. I man avanages ae ha, s nuve an can be smply fomulae, explc, uncononally sable snce s a passve newo whch s solve exacl can be use o moel abay an complex sucues, nhomogeneous mea can be moele vey convenenl an he mpulse esponse an me oman pefomance of he sysem can be oban saghfowaly. TLM meho s use o moel self-heang n vaous wo eleconc evces an sucues: AlGaN/GaN powe anssos an nsulae gae bpola ansso (IGBT) moules. esuls show ha he meho s well sue fo unesanng hea managemen n mcoeleconc evces an gves nsghs fo fuue esgns. Key wos TLM ; Moelng ; Self-heang ; powe anssos ; IGBT moules 1. Inoucon Tansmsson-Lne-Max (TLM) meho, ognally evelope n 1971 as a numecal echnque fo moelng elecomagnec wave popagaon [1], has snce been esablshe as a poweful echnque o suy ffuson poblems [2-4], vbaon [5,6], hea ansfe [7,8], elecomagnec compably [9], aa [10], ec. I s base on Huygens pncple an coul be use fo moelng any phenomena whch obeys hs pncple. I s a me an space scee meho ha solves fel poblems usng he eleccal ccu equvalen. I assembles a lace of scee pons n space connece by ansmsson lnes an efnes he ansmsson max beween lace pons, so ha successve calculaons can be pefome. Tansmsson-lnes ae consee as sbue moels of capacos, nucos an essos. The use heefoe has a goo gasp of he popees an behavo of he moel, he naue an sgnfcance of eos an he manne n whch maeal popees may be nouce. The physcal vaable s moele as a sequence of volage pulses avellng hough hs newo of ansmsson lnes o become ncen smulaneously on all pas of all noes. These ncen pulses ae scaee nsananeously no eflece pulses whch, ung he me sep Δ, avel along ln ansmsson lnes o become ncen upon neghbong noes. The TLM oune opeaes on he avellng, scaeng, an connecng of hese pulses n he newo. The ansmsson lnes n he moel ac as elay lnes, wh he noe mpulse populaon beng he scee soluon a each me sep. TLM s a scee moel whch can be solve exacly snce appoxmaons ae only nouce a he scesaon sage. Ths s o be conase wh he aonal appoach n whch an ealze connuous moel s fs obane an hen hs moel s solve appoxmaely. I man avanages ae ha, s nuve an can be smply fomulae, explc, uncononally sable snce s a passve newo whch s solve exacl can be use o moel abay an complex sucues. Inhomogeneous mea can be moele vey convenenl an he mpulse esponse an me oman pefomance of he sysem can be oban saghfowaly. In hs communcaon, he TLM meho s explane an use o moel self-heang n vaous eleconc evces an sucues. esuls show ha he meho s well sue fo unesanng hea managemen n mcoeleconc evces an gves nsghs fo fuue evces esgns. 2. TLM moel To oban se n empeaue of evce sucue, we solve he hea-flow nonlnea equaon: T ( T) T H( x, C (1) p T whee T(x, z, ) s empeaue funcon an H(x, z, ) s hea geneaon ae pe un volume; ρ s ensy of sol; Cp s specfc hea an s hemal conucvy. Ths ffeenal equaon s sceze when a physcal pocess s smulae wh a convenonal numecal meho. Ths s no he case wh TLM meho, whch s nnscally a

2 Tansmsson Lne Max: A Tool fo Moelng Themo-Elec JNTM(2012) A.Saane e al scee appoach ha moels a physcal pocess ecly. Is algohm peseves physcal sgnfcance of phenomenon une nvesgaon [2]. I s nown ha a pulse on a ansmsson lne obeys Maxwell s cul equaons fo popagaon n a lossy meum, whch leas o he elegaphe s equaon [4]: 2 2 C L C ² ² (2) whee Φ(x, z, ) s poenal an L, C an ae nucance, capacance, an essance pe un of lengh, especvely. Consan α = 1, 2 o 3 fo one, wo an hee mensons. Equaon (2) s whou loss f =0 an becomes consequenly encal o a wave equaon. On he ohe han, f 0, hen he equaon escbes an aenuae wave. When he fs me-eve em on he gh-han se of equaon (2) omnaes he secon me-eve em, one can esablsh equvalence wh he hemal ffuson pocess escbe by equaon (1). Hence, one paccal applcaon of he TLM meho s s use o moel physcal poblems whee eleccal/hemal analoges ae possble. The TLM meho opeaes on a g sucue an eplaces each noe wh an equvalen ccu of ansmsson lnes. An equvalen TLM noe of he sucue une nvesgaon s shown n Fg.1. When a pulse s sen on a ansmsson lne elemen, he poenal F obeys he elegaphe s equaon (2). In he TLM, aa n he fom of poenal o cuen values ae epesene by ela pulses. I s also suppose ha all hese pulses move n he newo n synchonsm. In Fg.1, he cuen geneao I s use o moel he geneaon of hea whn he evce. I epesens he powe nece n each noe (x, n he acve aea. Eleccal essance an localze capacance, n a ansmsson lne whou loss, moel hemal essance an hea capacy of he maeal, especvely. They ae gven by he followng equaons: x 2K yz, C xyz p Y y 2K xz, z 2K yx (3) An eave meho s use o oban noal poenal (empeaue) esulng fom he necon of ela pulses (npu hea) n he newo afe each me sep. Inee, when pulses avel own he ansmsson lnes, hey ae scaee a noes, wh pa of pulse eflece no he same lne an he ohe pa ansme o neghbong noes. Hence, local noes poenals pove an exac soluon o equaon (2) a each me sep. Decons of pulses on each numbee po ae shown on Fg.1. These pulses ae labele, wh ncang he econ an he h eaon. Applyng Thevenn an Mllman heoems o equvalen elecc ccu of a TLM noe, noal poenal a he h eaon can be eve: Y 1 Y Y 1 I Y (4)ICLE whee =3Δ/C s he chaacesc mpeance. eflece pulses a a noe, ncae by supescp, ae calculae accong o: 1,2 5,6 3,4 z z y y 1,2 3,4 5,6 Each pulse aes a me Δ o move own a ansmsson lne onng wo neghbong noes. eflecons a sconnues, ue o mpeance msmach, gve ncen pulses a (+1) h eaon: 1 ( x, ( x, 1 ( u, wh ' ( u,,, ( x, ( u, ( x, ( u, ( x, (5) (6) Table 1 shows,, u, v an w values. u v w 1 2 x-1 y z 2 1 x+1 y z 3 4 x y-1 z 4 3 x y+1 z 5 6 x y z x y z+1 Table 1 :,, u, v an w values. Implemenaon of a TLM oune consss solely of epeae applcaon of equaons (4) (6). TLM smulaon algohm can be acceleae by mesep changes an g coasenng. Fg.1. A sngle hee-mensonal TLM noe. 8

3 Tansmsson Lne Max: A Tool fo Moelng Themo-Elec JNTM(2012) A.Saane e al Fg.2 TLM bounay conons. Bounay conons expess neacon of sysem a han wh s suounngs. Bounaes ae pa of he anspo moel an hus shoul be conssen wh he escpon of hea anspo nse he meum. A hea snng bounay coespons o an eleccal sho-ccu (S/C) an Γ = -1, hus any ncen pulse on he bounay wll be eune equal n magnue bu evese phase. A hea nsulang bounay coespons o an eleccal open-ccu (O/C) an Γ = 1, hus any ncen pulse on he bounay wll be eune equal n magnue an n phase. A symmey bounay s epesene by an eleccal open-ccu n TLM moelng [2,4]. Convecve o aave hea losses a bounaes may be consee as a hea sn an moele usng an appopae eflecon coeffcen, Fg.2. A consan hea souce bouna Φb, s moele by ang, a he bounay neface: b b b whee b an b ae ncen an eflece pulses of he noe aache o he souce a he suface. Fg.3 Tempeaue sbuon aoun acve egon. Fg.3a shows he suface empeaue of GaN ha conans he acve zone. A ho-aea wh a maxmum empeaue of appoxmaely C, locae beween he souce an he an meaows of he HFET gae, s clealy vsble. Fg.3b s a 3D epesenaon of he same suface. I shows a osal-le empeaue sbuon wh a pea value nea he an because of he ssymmey. Wh hs epesenaon of TLM esuls, one can easly evaluae empeaue a each pon of evce. Ths epesens a gea avanage compae wh cean measuemen echnques such as, nfae (I) hemogaphy o mco-aman specoscop whee he empeaue-measue value s an aveage aen a a cean hcness of evce. 3. Self-heang n AlGaN/GaN powe anssos Gallum ne (GaN) an s compouns ae beng esablshe as maeals of exeme sgnfcance fo nex geneaon hgh-ensy powe evces. Due o he hgh beaown volages, hgh elecon mobl an hgh hemal conucves, hese maeals allow he fabcaon of evces, whch can opeae a volages, an empeaue anges beyon he convenonal semconuco maeals, such as GaAs an S. They ae use n mcowave aa an moble elecommuncaons [10,11]. AlGaN/GaN heeosucue fel-effec anssos symbolze by HFET, opeae a hgh fequency wh a hgh sgnal-o-nose ao. Howeve, opeaon of hese evces s lme by self-heang une pulse wong conons an ohe poblems assocae wh hgh enses of efecs. Ths self-heang maes hemal managemen a cucal aspec n evce-pacagng esgn. Seveal sysem-level coolng schemes have been pesene fo effcen hea sspaon, such as flp-chp bonng, wo-phase spay coolng an mounng evces ono hgh hemal conucvy subsaes [12 16]. Fg.4 Tempeaue sbuon along he z-axs. 9

4 Tansmsson Lne Max: A Tool fo Moelng Themo-Elec JNTM(2012) A.Saane e al Fg.5 Tempeaue sbuon a vaous eph levels. Fg.4 shows empeaue sbuon along he z-axs an Fg.5 shows empeaue sbuon along he x-axs a fou neface posons: AlGaN laye, AlGaN/GaN, GaN/SC an SC/sn. Ths vecal cu hough ffeen evce layes gves n-eph nfomaon on hea managemen n each laye. Fuhemoe, change n slope ncaes a change n hea ansfe ynamcs gong fom a laye o anohe. Tempeaue n he AlGaN laye of he evce s hghe han ha foun n he GaN laye [17,18]. Bu hea oes no accumulae a vaous nefaces, whch ohewse woul be a poblem snce mgh lea o ho-spo fomaon. Hence, n compason wh ohe analyss mehos, wh TLM one can eemne he place an fom of ho aea wh moe pecson. 4. Eleco-hemal analyss of IGBT moules. Tens n powe convee sysems ae o euce convee sze an o ncease swchng fequency. Boh equemens ncease hea geneaon aes whch become a ccal ssue o suy an opmze. Ths s he case of nsulae gae bpola ansso (IGBT) moules wh he hgh volage (seveal ) an hgh cuen (seveal A). Hee, a hemal analyss of a 1200 A, 3.3 IGBT moule was nvesgae an analyze usng Thee-Dmensonal Tansmsson Lne Max (3D- TLM), Fg.6, nclung measuemen an numecal esuls obane by ohe smulaos ools as MSC.PATAN [19], FLOTHEM [20] an LAASTHEM [21]. Themal smulaons wee un fo vaous powe loas an evces wh ffeen hea speaes such as AlSC, Cu Mo, Gaphe Cu an Cu of vaous hcnesses. IGBT moules have exhbe ffeen hemal behavos epenng on subsae maeal use. Maeals such as Alumnum Ne (AlN), Damon an BeO have paculaly goo hemal conucvy. These ceamcs pove eleccal nsulaon o he unelyng base plae, whch s usually pessue moune ono a hea sn by pepheal bols [22]. The use of ceamc subsaes s o ecease selfheang fom hea souces localze whn IGBT moules. Fg.6 Typcal TLM empeaue sbuon n an IGBT moule. Pecon of empeaue se s mpoan n pulse opeaon conons, Fg.7. Ths empeaue pofle s obane wh 100 W-sep pulses whee conol sgnals ae geneae va a PWM scheme. Tempeaue ses ae calculae usng TLM fo hee loa cycles an esul s shown n Fg.8. Dung fs loa cycle, empeaue se an op ae mpoan, an hen hemal gaen ecease wh nceasng loa cycles. Ohe smulaons wee pefome o show he mpoance of hea speaes ha ae nclue beween he evce an s hea sn. Fg.9 shows esuls fo hee hea speaes AlSC, Cu an CuMo. Themal ansens coss-ove occus a ffeen pons n me. The man ffeence beween Cu an CuMo n ems of hemal popees s ha Cu has a sgnfcanly hghe hemal conucvy K an a hghe hemal ffusvy han CuMo. Fg.7 Tempeaue evoluon n pulse moe. 10

5 Tansmsson Lne Max: A Tool fo Moelng Themo-Elec JNTM(2012) A.Saane e al Fg.8 Compason of TLM an MSC-NASTAN. empeaue eemnaon an pecse ho-aea sze nowlege ae vey ffcul o assess n boh HFET evces employng I mco-hemogaphy o n IGBT nefaces. The TLM can be a goo alenave esgn ool, snce allows he evaluaon of empeaue a any pon of he sucue. Is physcal esoluon epens on he g use only an complex geomees can be eal wh easly. Some TLM smulaons esuls have been compae wh hose obane expemenally usng negae mco-aman/i hemogaphy mehos o wh ohe moelng mehos, an have been foun o agee whn he bouns se by he esoluon eque. TLM meho wll pove hany n he suy of hea managemen of many maeals use n semconuco evces fabcaon. TLM has also he avanage upon ohe numecal mehos of beng uncononally sable, one sep, can aap o complex geomees, an smple o pogam. efeences Fg.9 TLM esuls fo 3 ffeen hea speaes. Tempeaue ffeences ae lage n he case of AlSC speae, whch s assocae wh hghe chp an sole empeaues. Alhough hemal ffusvy of AlSC falls beween ha of Cu an ha of CuMo, an s hemal conucvy s less han a half of ha of Cu an s specfc hea capacy - ensy pouc s only a h ha of CuMo, chp empeaues wee foun o be lowes wh a Cu speae an hghes wh an AlSC speae wh CuMo yelng nemeae values. Ths means ha he moel wh he CuMo hea speae always sas each successve pulse a a lowe empeaue han he moel wh a Cu hea speae. Snce he assembly s a layee sucue, maeals wh ffeen CTE can cause he layes of he assembly o wap when subece o ap hemal excaon. Theefoe, componen esgn shoul no only ae no accoun hemal an physcal poblems nheen o chp-o-hea sn assembl bu also neacon ssues beween layes of hea sn assembly self. Ths mples ha, as he numbe of pulses nceases, coolng aes of hea sn assembly wll play a moe mpoan ole n ensung an accepable oveall empeaue, as elavely moe hea wll be sspae wh each successve pulse. 5. Concluson The TLM meho was apple o suy self-heang n AlGaN/GaN evce sucues an nsulae gae bpola ansso (IGBT) moules. The meho s easy o pogam an gves nsghs on empeaue sbuon houghou he evce. I allows a bee unesanng of hea behavo an managemen a each laye ha foms he sucue. Absolue [1] P.B. Johns,.L. Beule, Poceengs of he IEE 118 (1971) [2] D. e Cogan, W.J. O Conno, S. Pulo, Tansmsson Lne Max Moellng n Compuaonal Mechancs, Taylo & Fancs, CC Pess, Lonon, [3] C. Chsopoulos, The Tansmsson-Lne Moelng Meho (TLM), IEEE Pess, New Yo, [4] D. De Cogan, Tansmsson Lne Max (TLM) Technques fo Dffuson Applcaons, Goon an Beach, [5] A. Saleh, P. Blanchfel,, Inenaonal Jounal of Numecal Moelng, 3 (1990) 39. [6] I.J.G. Sco, D. e Cogan, Jounal of Soun an baon, 311 (2008) [7] A.Am, A.Saane, S.Pulo, Compues n Bology & Mecne 41 (2011) 76. [8] S.Aloua-Bella, A.Saane, A.Hamou, M.Benzoha, J.M.Sae, Buns 34 (2008) 688. [9] W.J.. Hoefe, The ansmsson lne max (TLM) meho, n: T. Ioh (E.), Numecal Technques fo Mcowave an Mllmee-Wave Passve Sucues, Wle [9] A.K. Iye, G.. Elefheaes, Apple Physcs Lees 92 (2008) 131. [10] S. Nunc, E. Gebaa, J. Lasa, M. Has, IEEE Mcow. Mag. 3/1 (2002) 80. [11] M. Kuzuhaa, H. Myamoo, Y. Ano, T. Inoue, Y. Oamoo, T. Naayama, Phys. Sa. Sol. A 200/1 (2003) 161. [12] J.J. u, Y.F. Wu, S. Kelle, G. Pash, S. Heman, B.J. Thbeaul, U.K. Msha,.A. Yo, IEEE Mcow. Gue Wave Le. 9/7 (1999) 277. [13] J. Sun, H. Fama, A. Kouymo A. Chns,. Hu, H.M. Wang, J. hang, G. Smn, J. Yang, M.A. Khan, IEEE Elecon. Devce Le. 24/4 (1999) 375. [14] S.L. Lee,.H. Yang, Y. Hsyua, J. Hea Tansf. 116 (1994) 167. [15] S.Heffngon, W.. Blac, A. Gleze., n: Poceengs of Inenaonal Eleconc Pacagng Confeence, (2001) 779. [16] J.Y Muh C.H Amon, K. Gabel, P. Kuma, S.C. Yao,., n: Poceengs of Inenaonal Eleconc Pacagng Confeence, (2001)

6 Tansmsson Lne Max: A Tool fo Moelng Themo-Elec JNTM(2012) A.Saane e al [17] S. aasngam, J.W. Pomeo M. Kuball, M.J. Uen, T. Man, D.C. Hebe, K.P. Hlon,.S. Balme, IEEE Elecon. Devce Le. 25 (7) (2004) 456. [18] I. Ahma,. Kassomayaula, M. Holz, J.M. Beg, S.. Kuz, C.P. Tgges, A.A. Alleman, A.G. Baca, Appl. Phys. Le. 86 (17) (2005) 503. [19] MSC.PATAN, complee sofwae envonmen fo smulaon, (hp: / / ). [20] ANSYS FLOTHEM, Ansys Inc, (hp: / / [21] A. Ham, G. Coquez,. Lalleman, P. ales, Mcoelecon. elably J., 38 (1998) [22] C.an Gobol,. Sanaan, J.L. Hugns, IEEE Tans. Powe Elecon., 12 /1 (1997) 3 12