������������������ �������������������������� �!�������"��#�$ %�����&('�����������)�*���',+��"-�������!���"��&('���%��!�����/.�����0��!'����213'�465�����47���

Abstract8!9�:<;>=<?A@>BC@>DFE�GHDI@>E!DC?JDF9>:KBF9LBFM>:<N>OPBC:<=RQCSUTVE!9>N>W<X>=RY�Z>BF?JDFXO[N>X>DFW�N>E�X>DFE\E!DFX>M>QC:R=<N>96N>]^B�_>`aDFW<DFQC:RE�N>Y!:<;>DFE!O[BCW�O[N>X>DFW�]!N>E\B9>NcbdDFW�?eDC9>?JN>E^X>Dfb>=<QFD�gfhiN>X>DFW�N>E!X>DFE\E!DCX>M>QF:<=<N>9A=<?DF?J?JDC9>:R=<BCW�]!N>EBFQC;>=RDfb>=<9>jlk>M>=<QFm>W<nlDfbcBFW<M>BFZ>W<DFSdn>DC:"?J:<=<W<WoBFQFQCM>E!BF:<DpOPBCQFE!N>O[N>X>DCWN>]�:<;>D/X>Dfb>=<QFDCS[9>DFDFX>DCXq]!N>E�?Jn>?J:<DFO[Y!W<DfbcDCW ?J=<O[M>W<BF:<=<N>9^g�r�D@>E!DC?eDC9>:PZ>DCWRNcGs9>M>OPDCE!=<QFBCWP?J=<O[M>WRBC:<=RN>9(E!DC?JM>WR:<?iN>]2:R;>D3]�M>WRW<Y?JQFBCWRDVtu9>=<:<DVDFW<DFO[DF9>:vO[N>X>DFWvBF9>X[:<;>DVQCN>OP@>BCQF:<Y!E�DFX>M>QFDCXwN>E!X>DFEO[N>X>DFW<S�BF9>Xw?e;>NcGx;>N7Gx:<;>DV9>N>9>W<=<9>DFBFE�=R:<=<DF?y=<9[:<;>DV=<9>@>M>:v]!M>9>QFY:<=RN>9AQCBF9AZ>D�:RE�DFBC:RDCXADfbcDC9AGH=<:R;AW<=<9>DFBCE^E!DCX>M>QF:<=<N>9ABFW<j>N>E!=<:<;>O�g 1. Introductionz{E�DF?JDF9>:<DFX|X>Dfb>=<QFD}=<?~B}9>NcbdDFWlO[DF:<BCWlN>�>=RX>D}W<NcG�@>NcGHDFEO[=RQCE!N>;>N>:<@>W<BF:<D6j>BF?�?eDC9>?JN>EKBFE!E�BFn,�!�f��gd8!9�N>E!X>DCEK:<NI?J=<O[M>W<:RBC9>DFYN>M>?JW<n~BC?J?JM>E!D BIE!N>ZcM>?J:)X>DC?J=Rj>9�BC9>X�Df�>QCDFW<W<DF9>::<;>DFE!O[BCW)=<?JN>W<BFY:<=RN>9�N>]I:<;>D�OPDCO[Z>E!BF9>D�]!E�N>O�:R;>D�?JM>E!E!N>M>9>X>=<9>j�G�BF]!DCE�SV:<;>D?J=<WR=<QCN>9VOP=<QCE!N>;>N>:<@>WRBC:<D�@>W<BF:<]!N>E�Ox=R?�?JM>@>@>N>E!:<DCXHZ>nlj>WRBC?J?U@>=<W<W<BFE�?DFO[BF9>BC:<=R9>jP]!E!N>O�B�j>WRBC?J?KQFBF@6BCZ>NcbdD�:<;>D�?J=RW<=<QFN>96GHBF]!DCE\�u=Rj^g>�>g� W<BF?J?H�!zun>E!D��2�>�>�>�>�>��G�BF?HQF;>N>?JDF92]!N>E�=<:<?�W<NcG�:R;>DCE!O[BFWUQFN>9>YX>M>QF:<=�b>=<:<nABF9>XABC9>N>X>=<QpZ>N>9>X>BFZ>=<W<=<:<nA:RNA?J=<W<=<QFN>9^g

`HDF?J=<j>9>=<9>jaB�O[=RQCE!N>;>N>:<@>W<BF:<D�jdBC?L?JDF9>?JN>E�:R;>DCE!O[BFW6O[BF9>BCj>DFYO[DF9>:^=R?pN>])QCE!M>QF=<BCWv=<O[@>N>E!:<BF9>QCDog7�HDFE�DFZ>n ?JDfbcDCE!BFW^@>N>=<9>:R?p;>B�bcD:<NHZ>D�QCN>9>?J=RX>DCE!DFX>Sc?JM>QF;VBF?�:<;>D�;>N>OPN>j>DC9>N>M>?U:<DFO[@>DCE!BF:<M>E!D�X>=<?JY:<E!=<ZcM>:<=RN>9[NcbdDFEU:<;>DHjdBC?�?JDF9>?J=<:<=�bdD�E�D�j>=RN>9>?JS�BF9>X[:R;>DHj>N>N>X[:R;>DCE!YO[BFWwX>DFQCN>M>@>W<=R9>j�Z>DF:<GHDFDC9x;>N>:<@>W<BF:<D�BC9>Xx?J=<WR=<QCN>9xE!=<O�gy�U;>Dk>M>BFW<=<:<n|N>],:R;>=<?�:R;>DCE!O[BFW�X>DCQFN>M>@>W<=<9>j|QFN>E�E!DFW<BC:RDC?�GH=<:<;|:<;>D@>N7GVDFEVQFN>9>?JM>O[@>:R=<N>9iBC9>Xi=<9iQFN>9f��M>9>QC:R=<N>9,GH=<:R;,:R;>D�:<;>DFE!O[BCWO[BF?J?�N>][:R;>D,;>N>:R@>W<BF:<D,GH=R:<;�:<;>Di;>DFBC:R=<9>j�:<=<OPD�gU`VM>Di:<N3:<;>Dj>W<BF?J?�?J:RE�M>QF:<M>E!DCS�X>=<E!DCQF:AO[DFBF?JM>E�DFO[DF9>:<?�N>] :<;>D}:<DFO[@>DFE�BF:<M>E!DX>=<?J:RE�=RZdM>:R=<N>9�N>9�:<;>D6?JDF9>?JN>EK@>W<BF:<]!N>E�O�BCE!D69>N>:�@>N>?J?J=RZ>W<D�gc��;>M>?

O[N>X>DFW<W<=<9>j�BC9>X�?J=<O[M>W<BF:<=<N>9�BCE!D}:<;>D}N>9>WRn/G�BCn�:<N/Dfb7BFW<M>BC:RD:<;>DF?JD�@>BFE!BCO[DF:<DFE!?7g�U;>D�;>DFBC:�QFN>9>X>M>QC:<=RN>9�GH=<:R;>=<9�:R;>D�;>N>:<@>W<BF:<D�=R?(X>DF?JQFE�=RZ>DCX:<;>E!N>M>j>;A:<;>Dp]�N>W<WRNcGH=<9>jAj>NcbdDFE�9>=R9>jADCk>M>BF:<=<N>9>?7 

¡c¢o£GH;>DCE!D =<?[:R;>D�:<;>DFE!O[BFW�QCN>9>X>M>QF:<=�b>=<:<n�S =R?[:<;>D2?J@>DCQF=<t{Q;>DCBF:^QFBF@>BCQF=<:<n�S =<?p:<;>DlO[BF?J?pX>DF9>?J=<:<n�S =R?p:<;>Dl:<DFO[@>DCE!BF:<M>E!DX>=<?J:<E!=<ZcM>:<=RN>9>S =R?K:R;>D�;>DFBC:�j>DF9>DFE�BF:<=<N>9AE!BC:RD��!¤JN>M>W<D�;>DFBF:<=<9>jA=<?:<BFmdDC9�BF?A:R;>D X>N>O[=<9>BF9>:;>DFBC:R=<9>j�O[DFQF;>BC9>=<?JOP��S =<?l:<;>D ?J@>BFY:<=<BFW<W<n2b7BCE!n>=<9>j2DFW<DCQF:<E!=<QPQCM>E!E!DC9>:UX>DF9>?J=<:Rn�bcDFQC:<N>E�BF9>X =<?H:R;>D?J@>DFQC=<t{QVDCW<DFQF:<E!=<QVQCN>9>X>M>QF:<=�b>=<:<n"g¥TH?J?JM>O[=<9>j[:<;>BF:v:<;>DV;>DFBC:uj>DF9>YDCE!BF:<=<N>9 =R?pM>9>=R]�N>E!O[W<nwX>=<?J:RE�=<ZcM>:RDCXwGH=<:<;>=<9w:<;>Dl;>DFBF:<=<9>jwBCE!DFB�!W<M>O[@>DFXlE!DF?J=<?J:<N>E!�!SdBF9>Xl:R;>BC:":<;>D�?Jn>?J:<DFOxO[BF:<E�=RQCDF?�BCE!Dy:<DFO[@>DFE!YBC:RM>E�D�=<9>X>DF@>DC9>X>DF9>:�BCE!N>M>9>X6:<;>D�GHN>E!m>=<9>j6@>N>=<9>:<S":<;>D�tu9>=<:RD�DFW<DFYO[DF9>:�Z>BC?eDCX�?e@>BC:<=RBCW�X>=R?JQCE!DF:<=<¦FBC:R=<N>9�N>]6���f�PWRDCBFX>?�:<N§B�W<BFE¨j>DW<=<9>DFBCE^©H`HªU??Jn>?J:<DFOqN>]^:<;>D�]!N>E!O� 

�!«>�GH;>DCE!D BCE!D�:<;>D

global heat conductivityandheatcapacitymatrix, arethetemper-ature(state),the load and the output vector respectivelyand is thedimensionof thesystem.Theelec-tric power , which is assumedto becompletelytrans-formedinto the heatingpower is the input to the system.TH?A:R;>D BFZ>NdbdDw9>M>O[Z>DFEpN>]pDFk>M>BF:<=<N>9>?l=<?l:<N>N~;>=<j>;~]�N>Ey?JM>Z>?JDCYk>M>DC9>:?Jn>?e:<DCOPY�W<D�bdDFWK?e=<O[M>W<BF:<=<N>9>SUO[N>X>DFWE�DFX>M>QF:<=<N>9�=<?A9>DCQFDC?eY?JBFE�n"gK8!9a:<;>D§]!N>W<W<N7GV=R9>jaGHD§@>E�DF?JDF9>:69>M>O[DFE!=<QCBFW6?e=<O[M>W<BF:<=<N>9E!DC?JM>W<:R?IN>]6:R;>DL]!M>W<W<Y!?JQFBCWRDLt{9>=<:<D�DCW<DFO[DF9>:�O[N>X>DFW�BF9>X§N>]6:R;>DE!DCX>M>QFDCX�N>E�X>DFEyO[N>X>DFW�g"rLDwD��>@>WRBC=<9�]!M>E�:R;>DCEy;>N7G¬:<;>DwX>DF@>DF9>YX>DC9>QFDIN>]�;>DFBC:RDCE�­ ?6E!DF?J=<?J:<=�b>=<:RnLN>9L:RDCO[@>DFE!BC:<M>E!DIQFBC9�Z>DI:<BFmdDC9=<9>:<N�BCQFQFN>M>9>:Z>nLZcM>=<WRX>=<9>j�BIQCN>OP@>BCQF:O[N>X>DCW®g�8!9�?JDCQF:<=<N>9>?A«BC9>Xl_A:R;>Dptu9>=<:RDpDCWRDCO[DF9>:oO[N>X>DCWoN>]^:R;>DpX>Dfb>=<QFDpBC9>Xl:<;>DpO[N>X>DFWN>E!X>DCEE!DCX>M>QF:<=<N>9Ib>=RBATVE!9>N>W<X>=^BCW<j>N>E!=<:R;>O�BFE�DAD��>@>W<BF=<9>DFX^gc¯uDFQFY:<=<N>9I� ?J;>NcGH?�=R9IX>DC:RBC=<W^;>NcGq:R;>=<?�W<=<9>DFBFEKE!DFX>M>QC:R=<N>9IBFW<j>N>E!=<:<;>OQCBF9[Z>DVM>?eDCX[]!N>E�:RE�DFBC:R=<9>j[W<=<O[=R:<DCX[9>N>9>W<=R9>DCBFE�?en>?J:<DCOP?7gC8�9[?JDFQFY:<=<N>9>?°ABF9>XA±A:<;>DpE�DF?JM>W<:R?BC9>XAQFN>9>QCWRM>?J=<N>9ABFE�Dp@>E!DC?JDF9>:<DFX^g 2. Finite Element Model�U;>DVtu9>=<:RDlDFW<DFO[DF9>:v�!�uªU��O[N>X>DFWvQFN>9>:<BC=R9>?y:<;>DV:<;>E!DCDVO[BF=<9QCN>OP@>N>9>DC9>:<?�?J=RW<=<QFN>9�E�=<OPS^?e=<W<=<QFN>9�;>N>:<@>W<BF:<DF?HBF9>X�j>W<BF?J?�?J:<E!M>QFY:<M>E!D�BF?6GVDFW<WKBF?6:<;>D�@>W<BF:<=<9>M>O²;>DFBC:RDCE�BC9>XL:<;>DI:<;>DCE!O[BFWKQFN>M>Y

��BCO[BFE!B�³�DCQF;>:<N>W<X�´¶µ�·¸S>¤J¹>Eºj>DF9A�H=<W<X>DF9>Z>E!BC9>X�´¶µ�·¸S>¤J¹>Eºj>DF9Ar�»>W<W<DF9>?J:<DF=<9�´R¼�·cBC9>XA¤JBF9 � gf½pN>E!b>=<9>m�´Rµ�·´Rµ�·¸8!hi�UªU½HS>¾H9>=¶bcDCE!?J=<:RnAN>]\�{E�DF=<ZcM>E¨j>S � DFN>Eºj>DF?½y»>;>W<DCE^THW<W<DFD[�f�>_>S>�>¿^�>�f�A�{E!DC=<ZcM>E¨j>S � DCE!O[BF9dn�Sª�Y!O[BF=<W�  Z>DCQF;>:<N>W<X>À§=<O[:RDCm^g X>D´¶¼�·¸8!9>?J:<=<:<M>:RD�]!N>E^zu;>n>?e=<QCBFW�hiDFBC?eM>E�DFO[DF9>:���DCQF;>9>=<k>M>DF?JS>�HDC=<X>DF9>;>N>]!?J:<E�gfÁ>S>�>¿^�>�f�A�{E�DF=<ZcM>E¨j>S � DFE!O[BF9dn

�{=<j^g>�6h�=<QCE!N>O[BF?J;>=<9>DFX�OPDC:<BFWUN>�>=<X>D[j>BF?H?JDC9>?JN>E�BCE!E!BCn^Â�³�N>:<Y:<N>O�b>=<DCG/�!W<DF]�:R��S>:<N>@Ab>=RDCG/�!E�=Rj>;>:<��g

∇ κ Ã∇( )• Ä ρ Å Æ∂∂ Ö+ Ç Ä È Éσ-----=,=

κ Ê Ëρ ÃÌ Í

σ

Ì

Ê Î ˙ Ï Î+ Ð Ñ Æ( )=Ò Ó Ô Ã⋅=Õ Ö × Ø Ù Ø∈, Ú Û

( ) Ü Ý, × Þ∈,

Þ � _ ¿ ° °= ß à

( )

@>W<=R9>j}Z>DC:<GHDFDC9};>N>:R@>W<BC:RD�BF9>X}?J=<W<=RQCN>9}E!=<Oáb>=RB�:<;>D~@>W<BF:<=<9>M>ODFW<DFQC:<E!N>X>DF?�BF9>X�:<;>D6X>=<DFW<DFQC:<E!=<Q6@>BF?J?J=�b>=<?JOâWRBCn>DFE�gd�U;>D6:<;>DFE!O[BCWQFN>M>@>W<=<9>jIb>=<BA:<;>DABCOPZ>=<DC9>:^BC=<EG�BF?�QCN>9>?J=RX>DCE!DCXI=R9I:<;>DAO[N>X>DCW:<N>N^gf�{=<j^gf«A?J;>N7GV?:R;>D��{ª�O[N>X>DCWoGV=R:<;>N>M>:�:<;>DpBCOPZ>=<DC9>:oBC=<E�g

ãHM>O[DFE�=RQCBFW�?J=<OPM>W<BC:R=<N>9§E�DF?JM>W<:pN>]6:<;>D�:<DFO[@>DFE�BF:<M>E!DLX>=R?J:<E!=<ZcM>Y:<=RN>9INcbdDFE:<;>DlQC;>=R@IBF9>XIN>9 :<;>Dl;>N>:<@>W<BF:<DABFE!DA?J;>N7GV9 =<9 �u=Rj^gf_BF9>XA�u=<j^gf�^g

hiBF=<9 ?J=<OPM>W<BC:R=<N>9 E�DF?JM>W<:<?pBFE!DCS{:<;>BF:^;>DFBC:<DFEBF9>X DFW<DCQF:<E!N>X>DC?X>N}9>N>:pDC?J?eDC9>:<=RBCW<WRn§=R9>äuM>DF9>QCD�:R;>D�O[BF�>=<O[BFWp:<DFO[@>DCE!BF:<M>E!D�N>]:<;>Dp;>N>:<@>W<BF:<DFS>ZcM>:�E!BC:<;>DFE^=<:<?:<DFO[@>DFE!BC:<M>E!DpX>=<?J:<E!=<ZcM>:<=<N>9^gåUDF:vM>?�O[DF9>:<=<N>9>S�:R;>BC:u]�N>E�:R;>=<?yX>Dfb>=RQCDVBVX>n>9>BFO[=<QV:<;>DFE!O[BCW

Z>DC;>Bfb>=<N>E6=R?�N>]6?J@>DFQC=RBCW�=<9>:<DFE!DC?J:<S�D�g\j^g�GH;>DF9�jdBC?IO[DFBC?JM>E!DFYO[DF9>:<?�=R9V:<DFO[@>DFE�BF:<M>E!D�@>M>W<?JDFXVO[N>X>D�BFE!DyE!DFk>M>=<E!DCX^gJ¯uM>QF;VO[DFBFY?JM>E!DCO[DF9>:<?§;>B�bcD�:RGHNqO[BC=R9�BCX>b7BF9>:<BCj>DF?7 lt{E!?J:<SwO[N>E!DaX>BC:RB=<9>]!N>E�OPBC:<=RN>96QCBF96Z>D�E!DFQCDF=�bdDFX>S"?J=R9>QCD�O[DFBC?eM>E�DFO[DF9>:<?KBFE!D�X>N>9>DBC:{X>=R]¨]�DFE!DC9>:{N>@>DFE!BC:<=RN>9[:RDCO[@>DFE!BC:<M>E!DF?�BF9>XP?JDFQCN>9>X>So:<;>D�DC9>DFEºj>nQCN>9>?eM>O[D�=R?W<NcGHDFE�g 3. Dynamic Compact Thermal Modeling viaMoment Matching8!9§:<;>D�W<BF?J:pn>DCBFE!?IO[BF9dn}@>BF@>DCE!?I;>BfbcD�Z>DCDF9§X>DfbcN>:RDCX}:<N§:R;>Dj>DC9>DFE!BC:R=<N>9VN>]uX>n>9>BCOP=<QyQFN>O[@>BFQC:>:<;>DFE�O[BFW"OPN>X>DCW<?¥g�TV9HD��>:<DC9>Y?J=�bdDVE!Dfb>=<DfGxQFBF9wZ>DV]!N>M>9>Xw=R9L��«>�fgCTxW<BFEºj>DV?eDC:uN>]�BF@>@>E�N>BFQF;>DC?=<?�Z>BF?JDFX�N>9�tu:<:<=R9>j�BC9�æ�ç�W<BFX>X>DFE�9>DF:<G�N>E!m�N>9�:<;>D6N>Z>?JDCE!bcDCX?Jn>?J:<DFOáE!DF?J@>N>9>?JD~M>?J=<9>j}B~?JM>=R:<BCZ>WRD�N>@>:R=<O[=<¦FBC:R=<N>9}BFW<j>N>E!=<:<;>O��_>�fgd¯uM>QF;�BP9>N>9>Y!BFM>:<N>O[BF:<=<QPBF@>@>E!N>BCQF;�E�DFk>M>=<E!DC?�:R;>DPX>DF?J=<j>9>DFE:<NIQF;>N>N>?JD6:<;>DAE!=<j>;>:\9>M>O[Z>DFEKBF9>XI@>N>?J=<:<=<N>9IN>]:<;>DAæ�çaW<BFX>X>DFEDCWRDCO[DF9>:<?pGH=<:<;>N>M>:^?J:RE�=RQC:^j>M>=<X>DFW<=<9>DF?JS{BC9>X :<N @>DFE!]�N>E!OèBA:R=<O[DFYQCN>9>?eM>O[=<9>j�@>BFE!BCO[DF:<E!=<¦FBC:<=RN>9�BF9>X�?J=<OPM>W<BC:R=<N>9�N>]�:<;>D�]!M>W<WRY?JQFBCW<DPO[N>X>DFW�g>TH9>N>:<;>DFEéS^OPN>E�DPBFM>:<N>O[BF:<=<QPGHBFn2:<N2@>E!N>QFDCDFX>S^=<?:<Nw@>DCE!]!N>E�O¬O[BF:<;>DFO[BF:<=<QFBCWvN>E!X>DCE)E!DFX>M>QC:R=<N>9 N>]�:<;>Dl©H`HªUY�?Jn>?eY:<DFO��!«>�fg"8�9>:RN~:<;>=<?Vj>E�N>M>@�Z>DCW<N>9>j>?eS�]�N>EyDf�>BCO[@>WRD BwQFN>O[O[DFE!YQC=RBCW<WRn�BfbcBF=<W<BFZ>W<DLOPN>X>BCW�E!DCX>M>QF:<=<N>9§BF@>@>E�N>BFQF;x���>�!SyZ>BF?JDFX�N>99>D�j>W<DFQC:R=<9>jèBx9>M>O[Z>DFE}N>]}9>N>9>Y!E!DCWRDfbcBF9>:~DC=<j>DF9¥bcBFW<M>DF?7gl�U;>DO[BF=<9VX>=<?eBCX>b7BC9>:<BFj>D�N>]u:R;>=<?�BF@>@>E!N>BCQF;V=<?U:<;>BF:>:<;>DCE!D�=<?UBCjdBF=<9V9>Nj>M>=<X>DFW<=<9>D,;>N7G�:<N�QC;>N>?eD,:<;>DiX>N>O[=<9>BF9>:VO[N>X>DF?7g�T�E!DFW<BC:RDCXBC@>@>E!N>BFQC;V:R;>BC:"GHD�@>E!N>@>N>?JD�;>DCE!D�=<?�B�O[N>O[DF9>:"O[BF:<QF;>=<9>jV:<DFQF;>Y9>=<k>M>Dyb>=RByTHE!9>N>W<X>="BCWRj>N>E�=R:<;>O��¶°>�!ScGH;>=<QF;lQFBF9lZ>D�X>DC?eQCE!=<Z>DFXlBF?]!N>W<W<NcGH?7g� N>BCW�=<?�:RNP:<E!BC9>?e]�N>E!O�:<;>D�DCk>M>BF:<=<N>96?Jn>?J:<DFO��!«>��=<9>:<N6:<;>D�?J=<O[=RYW<BFE\?Jn>?e:<DCO~ 

�!_>�GH=<:<;PO[M>QC;P?JO[BFW<W<DFEUX>=RO[DC9>?J=RN>9 gF�HDCE!D QCBF9PZ>DH?JDFDC9BC?)By@>E!Nf��DFQC:<=RN>9AN>]v:<;>DêY!X>=<O[DF9>?J=<N>9>BCW":<DCOP@>DCE!BF:<M>E�Dyb>DFQC:<N>Ev:<NY!X>=<O[DF9>?J=<N>9>BFW�?JM>Z>?J@>BFQCDFS>?JM>Zf��DFQC:<DFXA:<NA?JN>OPD�DFE�E!N>Eë 

�!�>�BC9>X =R?�:<;>N>?JDaWR=<9>DCBFELQFN>O[Z>=<9>BF:<=<N>9�N>]L:R;>DE!DCX>M>QFDCX�?J:RBC:<DF?VGH;>=<QF;~QFN>E!E!DC?J@>N>9>X>?V:<N�:R;>DwE!DCk>M>=RE�DFX�?J:<BF:<DF?�ì=<9ADFk>M>BC:<=RN>9A��«>��g�U;>D[O[BF:<E!=<� =R92���>��=<?HQCN>OP@>N>?JDCX2]!E!N>O Y!X>=<OPDC9>?J=<N>9>BFWbdDFQF:<N>E�?�:<;>BF:\]�N>E!O�B6Z>BF?J=<?�]!N>EK:<;>D6E!DFX>M>QCDFX�?eM>Z>?J@>BCQFD�gcr3;>DC9:<;>D6?JM>Z>?J@>BCQFDP=R?H]!N>M>9>X�:<;>D6GH;>N>W<DPDFk>M>BF:<=<N>9��!«>��=<?�@>E!Nf��DFQC:RDCXN>9>:<Nw=<:u��Z>n[BC@>@>W<n>=R9>jw���>��BF9>Xw:<;>DF9wO[M>W<:<=R@>W<n>=<9>jw�!«>��]�E!N>O¬:R;>DW<DF]�:?J=<X>DIZ>n �!S�BF9>XL:R;>=<?6@>E!Nf��DFQC:R=<N>9L@>E!N>QFDC?J?A@>E!N>X>M>QCDF?ABE!DCX>M>QFDCX~N>E�X>DFEp?Jn>?J:RDCO��!_>�pBCQFQCN>E!X>=<9>j~:<N�:R;>D z�BCX>D N>Epz�BFX>DFY:<n>@>DpBC@>@>E!N>�>=<O[BF:<=<N>92��±>�� 

�!°>�8!:�QFBF9�Z>D6?J;>N7GH9�:R;>BC:\:<;>DA:<E�BF9>?J]!DCE]�M>9>QF:<=<N>9>?�N>]K:R;>D6?Jn>?J:<DFO[?�!«>�\BF9>XA�!_>�\GH;>DF9AX>DfbdDFW<N>@>DFXA=<9>:<NAhiBFQCWRBCM>E!=<9AD��>@>BF9>?J=<N>9^ 

�{=<j^gf«A�u=<9>=R:<D�DFW<DFO[DF9>:�O[N>X>DFW�N>]\:R;>D�j>BF??JDF9>?JN>Efg

�{=<j^gf_Temperaturedistributionoverthechipfor appliedconstant

heating power of 340mW and initial temperature of .

�{=<j^gf�Temperaturedistribution on thehotplatefor appliedcon-

stantheatingpowerof 340mWandinitial temperatureof .

« �°Ö

« �°Ö

Ö í Ú î˙ Õ í Ú î+ Ü î ß à( )=

ì í Ý í ï ð í⋅=

ñ Þ«ð í

Þñ ε

Ú ò Ú îεÚ î × í ñ Þ«,∈,+⋅=ì í Ý í ï ð í

⋅=

ò ñ Þò ï

Ö í ò ï Ö ò Õ;í ò ï Õ ò Ü í;=

ò ï Ü Ý í;ò ï Ý= = =

�!±>�GH;>DFE�D =<?HQCBFW<W<DFX�:<;>D[= à óO[N>OPDC9>:<S>OPBC:<QF;A:<;>DptuE!?J:�E^O[N>O[DF9>:<?7g��N>E\:<;>D�WRBCE¨j>D�W<=R9>DCBFE�?Jn>?e:<DCOP?�B�:<E!BC9>?J]!N>E!O[BF:<=<N>96O[BF:<E!=<�QFBC96Z>D�DC]¨]!DFQC:<=�bcDCW<n6QCN>OP@>M>:<DCX6BC?KBC96N>E�:R;>N>j>N>9>BCW�Z>BC?J=R?�]!N>E�:<;>D½HE!n>W<NdbA?JM>Z>?e@>BCQFD�N>]^:<;>DpX>=<O[DF9>?J=<N>9 ñ g

�!�>�GH=<:<; gp�U;>=<?,O[BF:<E�=R��=<?}B�X>=<E!DFQC:N>M>:<@>M>:\N>]KTHE�9>N>WRX>=�BFW<j>N>E!=<:<;>O�gd�{M>E!:<;>DCE!O[N>E!DFSv=<9,��°>�KGHD6;>B�bcD?J;>N7GH9>S:<;>BF:�]!N>EH:<;>D�W<=<9>DFBCE���=R9i?JDF9>?JD2N>]�:<DFO[@>DFE�BF:<M>E!D2=<9>X>DFY@>DF9>X>DC9>:A?Jn>?J:<DFO�OPBC:<E!=<QFDC?e�IX>n>9>BCO[=RQ�:R;>DCE!O[BFW6OPN>X>DCW<?JSH:<;>D¯{=<9>j>W<DFY�8!9>@>M>:<Y!¯u=<9>j>WRDCY!©HM>:<@>M>:�?JDC:RM>@�]!N>E�:<;>DaTVE!9>N>W<X>=�BFW<j>N>YE!=<:<;>O¬=<?y?JM>]étuQF=<DC9>:v:<NwBF@>@>E!N>�>=<O[BF:<Dl9>N>:vN>9>W<nwBl?J=R9>j>W<DlN>M>:<@>M>:E!DC?e@>N>9>?JD6ZcM>:\BFW<?JNI:<;>DA:<E!BC9>?J=RDC9>:\:<;>DFE!O[BFW\E�DF?J@>N>9>?JDA=<9IBCWRW\:<;>Dt{9>=<:<D3DFW<DFO[DF9>:P9>N>X>DF?�N>]�:<;>D3X>Dfb>=RQCDog��HDC9>QFDFSl]�N>E�:<;>DFE!O[BCW@>E!N>Z>W<DFO[?�=<:\]!M>9>QC:<=RN>9>?�DfbdDF9

whenE is a unit matrix of thedimension

g>©HM>Ey9>DfG|?eN>]�:<G�BFE!Dw:<N>N>W^ôHõ ñJö>÷ Þ>ø ì ø �¶�>��QFN>9>Y?J:<E!M>QF:<?~:<;>D}DCk>M>BF:<=<N>9�?Jn>?J:<DFOù�!«>� X>=RE�DFQF:<W<n/N>M>:lN>] THãH¯uúH¯Z>=<9>BFE!naDFW<DFO[DF9>:AO[BC:RE�=<�/t{W<DCSHBF9>Xa@>DFE!]!N>E�O[?�THE!9>N>W<X>=ABFW<j>N>YE!=<:<;>Os�!BC?2X>DC?eQCE!=<Z>DFX�BFZ>NdbcDC�P:<N�QFE!DCBF:<DiBiWRNcG�X>=RO[DC9>?e=<N>9>BCW?Jn>?J:RDCO�g 4. Nonlinearities in the Input Function³�n[X>DF?J=<j>9>=<9>j[BV;>DFBC:<DFE�=R:v=R?y9>DFQFDC?J?JBFE!nw:RNwQFBCWRQCM>W<BF:<DV=R:�­ ?�DCWRDCQFY:<E!=<QFBCWuE�DF?J=<?J:R=¶b>=R:<n g¥¾H9>X>DCE�:<;>DVBF?J?JM>O[@>:<=RN>9w:<;>BF:v:<;>DVDCWRDCQF:<E!=<Q@>N7GVDFEV=<?[QCN>OP@>W<DC:RDCW<ni:<E!BC9>?J]!N>E!O[DFX,=<9>:<Ni:<;>D�;>DFBC:<=R9>j,@>NcGHDFEGHDpQCBF9A?JDC:® 

�!Á>�GH;>DFE�D =R?�BF@>@>W<=<DFX�bcN>W<:RBCj>Dogd�HDFE�DFZ>nFSu:<;>D6;>DFBC:<DFE!?�E!DC?e=<?JY:<=�b>=<:<nA=R?B�]!M>9>QF:<=<N>9AN>]^:<DFO[@>DCE!BF:<M>E!D�=<:R?JDCW<]� 

�!¿>�GH;>DFE�D =<?E�DF?J=<?J:R=¶b>=R:<n6BF: S>BF9>X BF9>X BCE!D�:R;>D�:<DFO[Y@>DFE�BF:<M>E!D�QCN>DF]¨t{QC=<DF9>:<?7g���N>E�B2@>WRBC:R=<9>M>Oû?JDF9>?JN>E�­ ?P;>DFBF:<DCE�=<:�=<?DF9>N>M>j>;[:<N[BF?J?JM>O[DHW<=<9>DFBFE�:RDCOP@>DCE!BC:RM>E�DHX>DF@>DC9>X>DF9>QCDH=<9PE�BF9>j>D]!E!N>O :<N S>=®gfD�g QFBF9AZ>D�9>D�j>WRDCQF:<DFX^g8!9PN>E!X>DCE�:<N6Z>DHBFZ>W<D�:<NPQFN>9>:<E!N>W{;>DFBC:<DFE!?�E!DC?J=R?J:<=�b>=<:<n�SoBC9>XP?JN6:<;>D:<DFO[@>DFE�BF:<M>E!DLN>]6:<;>D�;>N>:R@>W<BC:RDCSyDFk>M>BF:<=<N>9��!¿>�6O[M>?J:�Z>DL:<BFmdDF9=<9>:RN BFQFQCN>M>9>:�g¥�HDF9>QCDFS�BlW<=<9>DFBFE)DFk>M>BC:<=RN>9 ?en>?J:<DCO¬�!«>��QC;>BF9>j>DC?=<9>:RNA:<;>D�GHDFBCmA9>N>9>WR=<9>DCBFE^?Jn>?J:<DFO� 

���f�>�`HDF@>DC9>X>DF9>QCDwN>]�:<;>Dw;>DCBF:<DFE!?lE!DC?e=<?J:<=�b>=<:<n�N>9~:<DFO[@>DFE�BF:<M>E!DwBF9>XQFN>9>?JDCk>M>DF9>:^X>DC@>DF9>X>DC9>QFDAN>]:<;>DA;>DFBF:<=<9>jI@>N7GHDCE)N>9I:<DFO[@>DFE�BFY:<M>E!DpBCE!Dp?J;>NcGH9A=<9A�{=<j^gf°ABC9>XA�{=<j^gf±^g�U;>DHO[BF=<9[k>M>DF?J:<=<N>9P=<?�9>N7G  cQFBC9[GHDHE!DCX>M>QFDH:<;>DHDFk>M>BC:<=RN>9?Jn>?J:RDCO����f�>�^Z>nlM>?J=<9>jA:R;>Dp?JBFO[DpW<=<9>DFBFE^THE�9>N>WRX>=<S>BF?)X>DC?JQFE!=<Z>DFX

BCZ>NcbdDFü��U;>D�BC9>?JGHDFE�=<?�n>DF?l��Á>�!SoZ>DFQFBCM>?JD�X>M>D�:<N6:<;>D�]!BFQF:<So:<;>BF::<;>Dw9>N>9>W<=R9>DCBFEy=<9>@>M>:�:<DFE�O X>N>DC?e9^­¸:�:<BFm>D[@>BCE!:�=<9~:R;>DO[N>X>DFWuN>E!X>DFEUE!DCX>M>QF:<=<N>9>So=<:{=<?�@>N>?J?J=RZ>W<DH:<NPNdbcDFE�:<BFmdDH=<:{=R9>:<N[:R;>DE!DCX>M>QFDCXA?Jn>?e:<DCOPS>GV;>=RQC;AGH=<W<Wo:<;>DC9AZ>DpN>]^:<;>D�]!N>E!O� ���>�f�

N>E�S>Z>nABC@>@>W<n>=R9>jAB�@>E!Nf��DFQC:<=RN>9A���>�� ���f«>�

zuW<DFBF?JDy9>N>:<DFSc:<;>BC: N>9V:<;>D�E�=Rj>;>:o;>BF9>XV?J=<X>Dy=R9lDFk>M>BF:<=<N>9>?����f�>�BC9>Xl���>�f�vQCBF9l@>E�=R9>QC=<@>BFW<W<nlZ>DyGHE�=<:R:<DC9lBF?)BCE!Z>=<:RE�BFE!nAWR=<9>DFBCEvQCN>O[YZ>=<9>BF:<=<N>9AN>]^?J:<BF:<DC?  ���f_>�

8!9wQFBC?JDVGH;>DF9 S�;>DCBF:<=<9>j[@>NcGHDFE�=<?�BV]�M>9>QF:<=<N>9[N>]�BV?J=R9>Yj>W<D69>N>X>D6:<DCOP@>DCE!BF:<M>E�DFSvBF9>X�B6@>E!N>X>M>QC: N>9�:<;>D6E!=<j>;>:;>BC9>XV?J=<X>Dy=R9lDFk>M>BC:<=RN>9l���f«>�vX>DFtu9>DF?)BFX>X>=<:<=<N>9>BFW"N>M>:<@>M>:�g =<?:<;>DF9�B6?J=<9>j>W<D6E!N7G�N>]KB6@>E!Nf��DCQF:<=<N>9IO[BF:<E!=<� gd8!9IQCBF?JD6GH;>DC9S N>9~:<;>DwE�=<j>;>:);>BF9>X~?J=<X>Dw=<9~DFk>M>BC:<=RN>9����f«>�y=<?:<;>N>?JD6WR=<9>DFBCEQCN>O[Z>=R9>BC:<=RN>9�N>]X>=<]¨]!DFE�DF9>:\E!NcGH?�N>]KBA@>E�Nf�®DCQF:<=<N>9O[BF:<E!=<� S>GV;>=RQC;AQFN>E!E�DF?J@>N>9>X>?:<NA���f_>��g�U;>D�=RO[@>W<DFO[DC9>:RBC:<=RN>9}N>]l�f�f�>�lBF9>X}���f«>�l]�N>El:R;>D�jdBC?w?JDC9>?JN>EO[N>X>DFW�GH=<W<W�Z>DpX>=<?JQFM>?J?JDFXA=<9A:<;>Dp9>D��>:�QC;>BF@>:<DFE�g

ý ø( ){ }ô

{ } þ ø þþ ÿ=

∞

∑–=

ô{ } þ Ý ï Ö µ– Õ

–( )þ µ+( )– Ö µ– �

= ò

Õ í � �,{ } ø � ÷ Þ � � �

…�í µ– �

, , ,{ }=� Õ

–µ– Ö � Õ

–µ– Ü=,=

Þ

×

ß à( )

� ¼ à( )×--------------=

� à( )

× × ð( )

× ÿ �αð

βð ¼

…+ + +( )⋅= =× ÿ �°Ö

α β

�°Ö ° � �

°Ö

β

Ê Î ˙ Ï Î+ Ð Ñ Æ Ã,( ) Ð � É Æ( )� Ã( )--------------⋅= =

Ò Ó Ô Ã⋅=

�u=<j^gf°A�HDFBC:RDCE!?E!DC?J=R?J:<=<b>=<:RnABC?Bp]!M>9>QC:<=RN>9AN>]\:<DFO[@>DFE!BC:<M>E!Dog

�u=<j^gf±��HDCBF:<=<9>j�@>N>GHDCE�BF?ABI]�M>9>QF:<=<N>9�N>]�:<DFO[@>DFE!BC:<M>E!DFS�O[DFBFY?JM>E!DCXA]!N>E^:<;>D�QFN>9>?J:<BF9>:�=<9>@>M>:ob>N>W<:<BFj>D�N>]��f��� g

� � � � � � � � � � � � � � � � � �

�

� �

�

� ��� ��� �� !" #$

% & % % ' % % ( % % ) % % * % %' ) %' * %' + %' , %' - %' . %

/ 0 1 2 0 3 4 5 6 3 0 7 8 9 :

; <=>?@ ABC

ß à ð,( )

Ö í Ú î˙ Õ í Ú î+ Ü î ß à ð,( )=

ì í Ý í ï ð í⋅=

ò ï Ö ò Ú î˙ ò ï Õ ò Ú î+

ò ï Ü ß à ò ∗ ð⋅í

,( )=

ì í Ý ï ò ð í=ðð þ ð D þ ð þ⋅þ∑=

E �= ò

∗ ð⋅í ò

∗òE �>

ò∗ ð

⋅í

ò

5. Results

5.1 Linear Model8!9/�{=<j^gf�/BF9>X/�u=<j^gfÁ/:R;>D§:RE�BF9>?J=<DF9>:A?JN>WRM>:<=<N>9/BF9>X/DCE!E!N>EZ>DF:<GHDCDF9�:<;>D2]!M>W<W<Y!?JQFBCW<D�BC9>X�:<;>D2E!DFX>M>QCDFX�O[N>X>DCW�=R9iQFBC?eD2N>]QFN>9>?J:<BF9>:�;>DFBF:<=<9>jA@>NcGHDFE\N>]^_>�>�>O[r BFE!D�?J;>NcGH9^gf8!96:R;>=<?QCBF?JDFSB�@>M>E�DFW<nV:<;>DFE!O[BCW"?en>?J:<DFOxGHBF?�=<OP@>W<DCOPDC9>:<DFXV�!M>?J=<9>jV;>DCBF:"j>DF9>YDFE�BF:<=<N>9IE!BF:<D6BF?�BAW<N>BFX�=R9ITVãH¯{úV¯{�^g �>�KBF9>XIE�DFX>M>QFDCX^gc�HDC9>QFDFS:<;>D DFk>M>BF:<=<N>9>? �!«>�ëBF9>X �!_>�ëGHDCE!D =<9>:RD�j>E�BF:<DFX ]!N>Eg��U;>D�]!E!DCk>M>DF9>Q�nPE!DF?J@>N>9>?JDC?KN>]�:R;>D�N>E�=Rj>=<9>BCWO[N>X>DFW"BC9>XV:<;>E!DCDyE!DFX>M>QCDFXVN>E�X>DFEvO[N>X>DFW<?�BCE!Dy?J;>N7GH9l=<9V�u=Rj^gf¿^g8!:\:RDC?J:R=<tuDF?�BAGHDFW<W\BFj>E!DCDFO[DF9>:\Z>DF:<GHDFDC9 :<;>DA:<E!BC9>?J]!DFE]!M>9>QC:<=RN>9N>]�:<;>D3]!M>W<WRY�?JQFBFW<D3BF9>X�N>]2:R;>D3E!DCX>M>QFDFX�O[N>X>DCWP=R9�W<NcGHY!]!E�DFYk>M>DF9>Q�nûX>N>O[BC=R9>S§BF?�j>E!BC9>:<DFXû:<;>E!N>M>j>;û:<;>D�X>DfbcDCWRN>@>O[DC9>:BFE�N>M>9>X =R9ADCk>M>BF:<=<N>9A�!±>��g

åUBF?J:<W<n�S":<;>E!N>M>j>;6O[N>X>DCW�N>E�X>DFE�E!DFX>M>QC:R=<N>96:<;>D�QFN>O[@>M>:RBC:<=RN>9:<=RO[DLGHBF?�E!DFX>M>QCDFX�Z>n�BL]ºBFQC:<N>E6N>]��f�>S�=<9>QFW<M>X>=<9>j�:<;>DL:R=<O[D9>DFDCX>DFX�]�N>E�:R;>D�bdDFE!n�E!DFX>M>QC:R=<N>9�N>]L:<;>D�DFk>M>BC:R=<N>9�?Jn>?e:<DCO�{=<j^g>�f�^g5.2 Nonlinear Model8!9��u=<j^g>�>�6BF9>X~�{=<j^g>�f«~:R;>Dw:<E!BC9>?e=<DC9>:�?JN>WRM>:<=<N>9~BF9>X�DCE!E!N>EZ>DF:<GHDCDF9�:<;>D2]!M>W<W<Y!?JQFBCW<D�BC9>X�:<;>D2E!DFX>M>QCDFX�O[N>X>DCW�=R9iQFBC?eD2N>]:<DFO[@>DFE�BF:<M>E!D}X>DC@>DF9>X>BF9>:A;>DFBC:<=R9>j/@>NcGHDFE ��BFQFQCN>E!X>=<9>j/:<N�:<;>D�{=<j^gf±>�)BFE!Dl?J;>N7GV9^g¥THW<?JNw=<9w:R;>=<?yQCBF?JDFS{BV@>M>E!DCW<nw:<;>DFE!O[BFWv?Jn>?JY:<DFOqG�BF?=<OP@>W<DCOPDC9>:<DFXA�!M>?J=<9>jA:R;>D�9>N>9>WR=<9>DCBFE^;>DFBC:oj>DC9>DFE!BC:<=RN>9

E!BC:<D�BC?KB�:<BFZ>W<D�W<N>BFX6=<96THãH¯uúH¯u�^g �>�\BC9>X6E!DCX>M>QFDFX^g�T�:<DFO[@>DFE!YBC:RM>E�DyN>]v:<;>DyO[DFBC9>X>DFE^G�BF?);>N7GVDfbcDFE^BF@>@>E!N>�>=<O[BF:<DFXl:<;>E!N>M>j>;lB?J=<9>j>W<Da9>N>X>D/:<DFO[@>DCE!BF:<M>E!DCS[?JBCn S[?JNq:<;>BC:I:<;>D/;>DCBF:<=<9>j@>NcGHDFE\QFN>M>W<XAZ>DpD��>@>E!DC?J?JDFXABF?7 ���f�>�

�HDC9>QFDFSo:<;>D�DCk>M>BF:<=<N>9>?����f�>��BC9>XP���f«>��GHDCE!D�=<9>:<D�j>E!BC:RDCX>SoZ>nM>?J=<9>j2:<;>N>?JD[WR=<9>DCBFE�QFN>O[Z>=<9>BF:<=<N>92N>]�E�DFX>M>QFDCX2?J:<BF:<DF?=<9A���f«>��S>GH;>=<QF;AQFN>E�E!DF?J@>N>9>X>?:<N g

TH?�D��>@>DFQC:RDCX>S�:R;>DxE�DFBCQF;>DFX¬?J:<DFBCX>n>Y!?J:<BF:<DxN>],9>N>9>W<=<9>DFBFE:<;>DFE�OPBCW OPN>X>DCWwX>=<]¨]!DFE�?,]�E!N>O :R;>D�N>9>D/]!N>E~W<=R9>DCBFE~:<;>DFE�O[BFW

�{=<j^gf�Solutionof the full 73. 955 ordersystemandof the re-

ducedorder-50 systemin a centralhotplatenode,for constantheating power of 340mW.

�{=<j^gfÁ�`V=R]�]!DFE�DF9>QCD�Z>DF:<GHDCDF9i:R;>D2]!M>W<W<Y!?JQFBCW<D�BF9>Xi:R;>D2E!DCX>M>QFDFX?JN>W<M>:R=<N>9>S>QCN>E!E!DC?e@>N>9>X>=<9>jA:<NAE!DC?JM>WR:<?=<9A�u=Rj^gf�^g

ß à( )

_ � � ô F=

ø �=

G H I J K LGL GH G GH L GI G GI L GJ G G

M N O O P Q R S O T U V W T OX T W N R T W V Y W T Y U V W T O

Z [ U T \ Q ]

^ _`a_bcd eb_f ghi

j k l m n op k jp qp rp np lj

s t u v w x y

z {|}{~�� �~{� � ��

�u=<j^gf¿A�{E!DCk>M>DF9>QCnAE!DF?J@>N>9>?JD�g

�u=<j^g>�f�Computationaltimes for a full-scale model (ANSYS)

andthereducedmodel(MOR) showing a decreaseby a factorof10 (computations done with Ultra SPARC 10).

�u=<j^g>�>�Solutionof the full 73. 955 ordersystemandof the re-

ducedorder-50systemin acentralhotplatenode,for temperaturedependant heating power according to Fig.6.

� � � � � � � � � � � � � �

� � �� � �� � �� � �

� � � � � � � � � � � � �

�� ��

� � � � �   ¡ ¢ � £ ¤ ¥ ¦ £ �§ £ ¦ � ¡ £ ¦ ¤ ¥ ¦ £ � ¥ ¨ ¦ £ ¨ ©§ £ ¦ � ¡ £ ¦ ¤ ¥ ¦ £ � ¥ ¨ ¦ £ ¨ © ª§ £ ¦ � ¡ £ ¦ ¤ ¥ ¦ £ � ¥ ¨ ¦ £ ¨ « ª

¬­ ¬ ¬® ¬ ¬¯ ¬ ¬° ¬ ¬

± ² ³ ´ ³ µ ¶ ·

¸ ¹ º » ¼

½¾ ¿ÀÁ ÂÃ

Ä ÅÆ Ç ÈÉ Ê Ë ÌÍÎÏ Ð

ÑÒ ÎÓ

Ô Õ Ö × Ø Ù Ö Ú Û Ü× Ô Ý Þ Ù Ö Ú Û Üß Û à Þ Ö Ú Û Ü

ð áß à ð

,( )ß ð á à

( )( )=

ò∗ ð

⋅íð á

â ã ä å æ çâç âã â âã ç âä â âä ç â

è é ê ê ë ì í î ê ï ð ñ ò ï êó ï ò é í ï ò ñ ô ò ï ô ð ñ ò ï ê

õ ö ð ï ÷ ì ø

ù úûüúýþÿ �ýú�� ��

O[N>X>DFW<SwZcM>:�:R;>D�E!DFX>M>QCDFX�O[N>X>DFW�X>DF?JQFE�=<Z>DF?�:R;>N>?JD�?J:RDCBFX>n>Y?J:<BF:<DF?QFN>E�E!DFQC:<WRnA=<9AZ>N>:<;AQFBC?JDF?7g5.3 HDL ModelãHN>9>W<=R9>DCBFE�=R:<=<DF?3=<9�:<;>D(=R9>@>M>:2]�M>9>QF:<=<N>9�QFBC9�]!M>E!:<;>DCE�Z>D:<E!DFBC:<DFXlGV=R:<;>=<9ABp?en>?J:<DFOqWRDfbdDFW�OPN>X>DCW�g��u=Rj^g"�f_l?J;>N7GV?)Bp?J:RE�M>QFY:<M>E!DlN>]�Bl�H`Hå,O[N>X>DFWv]!N>E)jdBC?y?JDF9>?JN>E�g¥8!:vQFN>9>:<BC=R9>?pBVE!DCX>M>QFDFXN>E!X>DCEHO[N>X>DFW�=R9i]�N>E!O�N>]HBF9iDFk>M>BC:<=RN>9i?Jn>?J:<DFO��!:<;>=<?[=R?[?JM>@>Y@>N>E!:<DFX§Z>n§D�g\j^g\¯{TH³�ªUæ�?J=<OPM>W<BC:RN>E��AGH=<:<;§B�Z>BCQFm§QFN>M>@>W<DFX;>DFBC:RDCE!?[E!DF?J=<?J:<N>EHX>DF@>DF9>X>BC9>:�N>9iB2?e=<9>j>W<D29>N>X>D2:<DFO[@>DFE�BF:<M>E!DFSGH;>=<QF;A=<?Bp@>BCE!:<=<QFM>W<BFE\WR=<9>DCBFE^QCN>OPZ>=<9>BC:R=<N>9AN>]\E!DFX>M>QCDFXA?J:<BF:<DF?7 

���f°>��U;>DAN>M>:<@>M>:\9>N>X>D6:RDCO[@>DFE!BC:RM>E�D =<?�BCj>BF=<9IBAW<=<9>DFBCEQFN>O[YZ>=<9>BF:<=<N>9AN>]^E!DCX>M>QFDFXA?J:<BC:RDC?JS>X>DF]!=<9>DCXABF? =<9A���>�f��g³�BC?J=RQCBFW<W<n>S�:<;>DFE�D~=<? 9>N§X>=R]�]!DFE�DF9>QFD�Z>DF:<GHDCDF9§:R;>=<? =<O[@>W<DFYO[DF9>:<BF:<=<N>9ABF9>XA:<;>Dp=<O[@>W<DFO[DF9>:<BC:R=<N>9A=<9AQF;>BC@>:<DFE^°^g «^g

6. Conclusionr�D ;>BfbdDw?J;>NcGH9~:<;>BF:Z>n~O[DFBF9>?lN>]pO[N>X>DFW)N>E!X>DCEyE�DFX>M>QFY:<=<N>9/b>=<B§THE�9>N>W<X>=ABF9aBFQCQFM>E�BF:<D§X>n>9>BFO[=<Q§QFN>O[@>BFQC:A:<;>DFE�O[BFWO[N>X>DFW�N>]\:<;>D�O[=RQCE!N>;>N>:<@>W<BF:<D�jdBC?K?eDC9>?JN>E\QFBF96Z>D�j>DC9>DFE!BC:<DFX^g�8!:;>BC?2Z>DFDC93]!M>E�:<;>DFE[?J;>N7GH9>S�:R;>BC:H:<;>=<?2W<=<9>DFBFE[E!DCX>M>QF:<=<N>93BCW<j>N>YE!=<:<;>O²QCBF9LZ>D�GHDCWRW�M>?JDFXL]�N>E�W<=<O[=R:<DCXL9>N>9>W<=<9>DFBFE�?Jn>?J:<DFO[?JS�=<9QCBF?JD6GV;>DF9�:<;>D69>N>9>W<=<9>DFBFE�=R:<=<DF?�BC@>@>DFBFE�N>9>W<n�GH=R:<;>=<9�:<;>D6=<9>@>M>:]!M>9>QC:<=RN>9^g>��N>E�BPjdBC?�?JDF9>?JN>EéSv:<;>=<?H:<;>DPQFBC?eD[GH=<:R;2:<DFO[@>DCE!BF:<M>E!DX>DC@>DF9>X>DF9>QCDyN>]u:<;>D�;>DCBF:<DFE�?�E!DF?J=<?J:<=�b>=<:<nVN>Ev;>DFBF:<=<9>jV@>NcGHDFE�g��U;>D9>N>9>W<=<9>DFBCE!=<:R=<DC?KGH=<:<;>=<96:<;>D�=R9>@>M>:{]!M>9>QF:<=<N>96QCBF96BCW<?JN6Z>D�:RE�DFBC:RDCXX>M>E!=<9>jA:<;>Dp?Jn>?J:<DFOqW<DfbcDCW�?J=RO[M>W<BF:<=<N>9^g 7. Acknowledgments�U;>D THM>:<;>N>E!?lGHN>M>WRX�WR=<mdD :<N~:<;>BF9>m~`VE®g�ªUb>j>DF9>=<=)æ�M>X>9cn>=)]!N>E;>B�b>=R9>j~M>?JDFX�;>=<?V?JN>]!:<G�BCE!Dw:<N>N>W\ôHõ ñJö>÷ Þ>ø ì ø g"�U;>=<?lG�N>E!m�GHBF?@>BCE!:<=RBCW<WRnl]!M>9>X>DFXVZ>nV:<;>D�ªU¾�:<;>E!N>M>j>;V:<;>Dy@>E!Nf��DFQF: � W<BC?e? � BF?JY!8!8�!8�¯{�oY�¿>¿>Y��f¿>�>�>_>�PW<DCX3Z>n3`HE�gUçAgUç�BF9>DiBF9>X�@>BFE!:<=<BFW<W<n3Z>n3BC9N>@>DCE!BF:<=<9>jAj>E!BC9>:oN>]\:R;>D�¾H9>=�bdDFE�?e=<:<nAN>]^�uE!DC=RZcM>Eºj^g 8. References

[1] J. Wöllenstein,H. Böttner, J. A. Plaza,C. Cane,Y.Min, H. L. Tuller, “A novel single chip thin filmmetaloxidearray”,Sensors and Actuators B: Chem-ical 93 (1-3), pp. 350-355, (2003).

[2] M. N. Sabry, “Dynamic CompactThermalModelsUsedfor ElectronicDesign:A Review of ReacentProgress“,Proz. IPACK03, (2003).

[3] M. Renc,V. Szekely, “Dynamic ThermalMultiportModeling of IC Packages“,IEEE Trans. CPT, 24,pp. 596-604 (2001).���>� THãV¯{úH¯2æ�DFW<DCBF?JDH°^g ±P¾H?JDCE�hiBC9>M>BFW<?JY!�U;>DCN>E!nPhiBC9>M>BFW<S���f¿>¿>¿>�fg��°>� ��gC³�DFQF;>:<N>W<X>S�ª�gC³AgCæ�M>X>9cn>=<S�¤¥g � gC½pN>E!b>=<9>m>S�THM>:<N>OPBC:<=RQ� DF9>DCE!BF:<=<N>9�N>]Kç�N>O[@>BFQC:\ªUW<DCQF:<E!N>Y!��;>DFE!O[BFW�hiN>X>DCWR?�]!N>E¯{DCO[=RQCN>9>X>M>QF:<N>Ep`HDfb>=<QFDC?JS�� Ý ��Ö Ý ð ñ*÷ Þ>ø�� õ Þ Ý � D à ñ õ Þ�� SªUÁ>±>Y!ç�S� "S>@>@^gf�>°>¿>Y!�>±>°>S>�!«>�>�>_>�fg��±>� ªpg³lgæ�M>X>9dn>=<S�¤7g � g½yN>E�b>=<9>m>S�æ�Dfb>=RDfGI �THM>:<N>OPBC:<=RQhiN>X>DFW�æ�DCX>M>QF:<=<N>9 ]!N>E �vE�BF9>?J=<DF9>:è¯u=<OPM>W<BC:R=<N>9 N>]hiªUhi¯uY!Z>BF?JDCXL`HDfb>=<QFDC?JS�� � Þ>ø õ ñ ø � ��� ÷ à � S����S�@>@^g{_>Y_>_>S>�!«>�>�>«>�fg���>� ��gH³�DFQF;>:<N>W<X>S ªUb>j>DC9>=R=2³AgHæ�M>X>9cn>=<S ¤7g � gH½pN>E!b>=<9>m>S� ªU]étuQF=<DC9>:yO[N>X>DCWR=<9>j}BF9>X}?J=<O[M>WRBC:<=RN>9}N>]l_>`�DCW<DFQF:<E!N>Y:<;>DFE�OPBCWHOPN>X>DCW�]!N>E[B�@dn>E!N>:<DFQF;>9>=<QCBFWHOP=<QCE!N>:<;>E!M>?J:<DFE � SzuE!N>¦ogfzuNcGHDFE!hiªUhi¯u«>�>�>_^g

[8] Z. Bai et. al., “New Numerical TechniquesandTools in SUGARfor 3D MEMS Simulation“,Proz.MSM2001, pp. 31-34 (2001).

�{=<j^g>�f«I`H=<]�]!DFE�DF9>QFD6Z>DF:<GHDCDF9I:<;>DA]�M>WRW<Y!?JQCBFW<DABC9>XI:<;>DAE!DCX>M>QFDFX?JN>W<M>:R=<N>9>S>QCN>E!E!DC?e@>N>9>X>=<9>jA:<NAE!DC?JM>WR:<?=<9A�u=Rj^g>�>�>g

�{=<j^g>�f_HDL model structurecontaininglinear reducedmodel

and back coupled temperature dependant heater.

� � � � � �� �� �� �� �� ���

� � � � � !

" #$%#&'( )&#* +,-

ð á ò∗ ð

⋅í

= ð . / 0 ì í

12 3 4 5

6 78 9

: ; < = > < ?@ A B C

D E FG H I J K I L M N O P Q R ST UT V T WT X

Y Z [ \ ] [ ^ _ ` a b c d b e f g h i j k l m n

o p q r p s t u v w