Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyProperty 8 (Reciprocity): In the Laplace transform domains, forĈ± for any couple of situations P + 1 (s), T+ 1 (s), P− 1 (s),T − 1 (s) and P + 2 (s), T + 2 (s), P − 2 (s), T − 2 (s), it results in2.5DiscretizedReducedT +T1 (s)P + 2 (s)+T −T1 (s)P − 2 (s) == T +T2 (s)P + 1 (s)+T−T 2 (s)P − 1 (s)21.5z 11(t)z 22(t)The matrices defining the compact model Ĉ± can be computedalso when u + (r), u − (r) are numerically extimated fromthe discretization of the advection-diffusion equation.1z 21(t)V. MOMENT MATCHINGLet w(r,t) be the row vectors of the temperature risescorresponding to the velocity field v(r) and to the sourcedensities g(r)δ(t). Letw(r,s) be the Laplace transform ofw(r,t). Let us introduce matching points α r with r =1...l.Expanding w(r,s) in a Taylor series around α r it results inw(r,s)=+∞∑0pw p (r,α r )(s − α r ) p .By substituting this expansion into Eqs. (1), (2) it straightforwardlyfollows in Ωα r c(r)w 0 (r,α r )+∇·(−k(r)∇w 0 (r,α r ))++ c(r)v(r) ·∇w 0 (r,α r )=g(r)α r c(r)w p (r,α r )+∇·(−k(r)∇w p (r,α r ))++ c(r)v(r) ·∇w p (r,α r )=−c(r)w p−1 (r,α r )with conditions on ∂Ω−k(r) ∂w p∂ν (r,α r)+ 1 2 c(r)v ν(r)w p (r,α r )=h(r)w p (r,α r ).Let us assume that the basis functions in the vectorsu + (r) and u − (r) spam the space spanned by the elements ofw p (r,α r ) for p =0,...,k r and r =1,...,l.Inthiswaythecompact model Ĉ turns out to be a Padè type approximant ofthe thermal network C. This is proved by the following result.Let Ẑ(s) be the impedance matrix of the compact model Ĉ.Let us consider the moments Z p (α r ) and Ẑp(α r ) in the Taylorseries expansionsZ(s) =Ẑ(s) =+∞∑p0+∞∑0pZ p (α r )(s − α r ) p ,Ẑ p (α r )(s − α r ) p .The following multi-point moment matching properties hold.Theorem 1: It isZ p (α r )=Ẑp(α r )for p =0,...,k r and r =1,...,l.0.5z 12(t)010 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1t (s)Fig. 2. Responses of the thermal network modeling the conjugate heattransfer problem.By choosing the functions in u + (r) u − (r) in this wayfor a choice of the matching points α r and matching ordersk r , with r = 1...l, a compact model is obtained whichapproximates the responses of Ĉ± and Ĉ and preserves theirthermodynamic properties. This is obtained without computingtransient solutions of the advection-diffusion equation, butjust solving the advection-diffusion equation in the complexfrequency domain at the chosen matching points.In order to avoid ill-conditioning problems [4] the functionsin u + (r) and u − (r) are not taken equal to the chosen momentsw p (r,α r ) with p = 0,...,k r and r = 1,...,l. Insteadan Arnoldi-like algorithm can be repeated for each of then elements of g(r) and for each expansion point α r withr =1,...,l as in [4].VI. NUMERICAL RESULTSA simple example is considered, composed by a siliconsubstrate with a pair of independent heat sources cooled byan incrompressible fluid. It can be assumed as a propotypeof a micro-channel. Such problem has been modelled bythe advection-diffusion equation (1). A discretization of thisequation has been obtained by the finite difference method[5]. Using such discretized model approximations of u + (r)u − (r) have been obtained for the choice of matching pointsα r and matching orders k r established by the authors forheat conduction problems [4]. Using these approximations ofu + (r) and u − (r) the compact model Ĉ of the C thermalnetwork has been obtained. By an order l =16approximationa 0.1% approximation in the power step thermal response ofC has been obtained, as shwon in Fig. 2. From this figure,a significant difference between z 12 (t) and z 21 (t) is clearlyobserved. Thus no choice of heat exchange coefficients, in anymodel of the problem based on the heat diffusion equation,could reproduce the generated results since then z 12 (t) =z 21 (t).©EDA Publishing/THERMINIC 2008 56ISBN: 978-2-35500-008-9