Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, Italyassumed−k(r) ∂x∂ν (r,t)+1 2 c(r)v ν(r)x(r,t)=h(r)x(r,t) (2)in which ν(r) is the unit vector outward normal to ∂Ω at r,v ν (r) =v(r) · ν(r) and h(r) is an heat exchange coefficientwhich is assumed to be non-negative.Eqs. (1), (2) modelling conjugate heat transfer problemsreduce to the heat diffusion equation with Robin’s boundaryconditions when the velocity field is zero v(r) =0. Eq.(2)has a particular form which solely ensures a passivity property.Property 1 (Passivity): For each time interval t 1 ≤ t ≤ t 2it results in∫ t2∫W (t 2 ) ≤ W (t 1 )+ dt g(r,t)x(r,t) dr,t 1 ΩW (t) being the nonnegative quantity∫1W (t) =2 c(r)x2 (r,t) dr.ΩEqs. (1), (2) do not satisfy a reciprocity property [1].However let y(r,t) be the temperature rise due to the generatedpower density f(r,t) in presence of the opposite velocityfield −v(r). Then the following even-odd formulation of theadvection-diffusion equation can be givenc(r) ∂x+∂t (r,t)+∇·(−k(r)∇x+ (r,t))++ c(r)v(r) ·∇x − (r,t)=g + (r,t)c(r) ∂x−∂t (r,t)+∇·(−k(r)∇x− (r,t))++ c(r)v(r) ·∇x + (r,t)=g − (r,t)with boundary conditions− k(r) ∂x+∂ν (r,t)+1 2 c(r)v ν(r)x − (r,t)=h(r)x + (r,t)− k(r) ∂x−∂ν (r,t)+1 2 c(r)v ν(r)x + (r,t)=h(r)x − (r,t)in whichx + (r,t)=(x(r,t)+y(r,t))/2x − (r,t)=(x(r,t) − y(r,t))/2g + (r,t)=(g(r,t)+f(r,t))/2g − (r,t)=(g(r,t) − f(r,t))/2.Such even-odd formulation of Eqs. (1), (2) satisfy both apassivity property and a reciprocity property.Property 2 (Passivity): For each time interval t 1 ≤ t ≤ t 2it results in∫ t2W ± (t 2 ) ≤ W ± (t 1 )+ dtt∫1(g + (r,t)x + (r,t)+g − (r,t)x − (r,t)) dr,ΩW ± (t) being the nonnegative quantity∫W ± 1(t) =2 c(r)(x+2 (r,t)+x −2 (r,t)) dr.ΩProperty 3 (Reciprocity): In the Laplace transform domains, for any couple of situations g 1 + (r,s), g− 1 (r,s), x+ 1 (r,s),x − 1 (r,s) and g+ 2 (r,s), g− 2 (r,s), x+ 2 (r,s), x− 2 (r,s) it results in∫(g 1 + (r,s)x+ 2 (r,s)+x− 1 (r,s)g− 2 (r,s)) dr =Ω∫= (g 2 + (r,s)x+ 1 (r,s)+x− 2 (r,s)g− 1 (r,s)) drΩIt is noted that if only situations in which g(r,s)=f(r,s)are considered such reciprocity property reduces to∫∫g 1 (r,s)x + 2 (r,s) dr = g 2 (r,s)x + 1 (r,s) drΩwhich is obtained from the reciprocity property for the heatdiffusion problem by substituting each temperature rise withthe mean of the two temperature rises corresponding to a sameheat source and two opposite velocity fields.III. THERMAL NETWORKSA thermal network C is introduced for Eqs. (1), (2) bydefining its powers and temperature rises, exactly as for theheat diffusion problem. The source term is supposed to begiven bybeingΩg(r,t)=g(r)P(t)g(r) =[g 1 (r),...,g n (r)],⎡ ⎤P 1 (t)⎢P(t) = ⎣ .P n (t)⎥⎦ .In this way the powers P 1 (t),...,P n (t) at the n ports of Care defined. The temperature rises⎡⎢T(t) = ⎣T 1 (t)..T n (t)⎤⎥⎦ .at the n ports of C are then defined by∫T(t) = g T (r)x(r,t) dr.ΩThe relation between port powers and temperatures is modelledby an n×n power impulse thermal response matrix z(t)and its Laplace transform, the n×n thermal impedance matrixZ(s).Such definition of thermal network preserves the passivityproperty 1 of Eqs. (1), (2). In fact©EDA Publishing/THERMINIC 2008 53ISBN: 978-2-35500-008-9