Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyProperty 4 (Passivity): For each time interval t 1 ≤ t ≤ t 2it results inP (t)1:1P + (t)∫ t2W (t 2 ) ≤ W (t 1 )+ T T (t)P(t)dtt 1T + (t)Equivalently, the thermal impedance matrix Z(s) is positivereal.A thermal network C ± can be introduced also for the evenoddformulation of the advection-diffusion equations. Thepowers P + 1 (t),...,P+ n (t) at n ports of C ± and the temperaturerises T − 1 (t),...,T− n (t) at the other n ports of C ± are definedbyT (t)1P − (t)T − (t)g + (r,t)=g + (r)P + (t)g − (r,t)=g − (r)T − (t)Fig. 1. The C thermal network synthesized by C ± and G.beingg + (r) =[g 1 + (r),...,g+ n (r)],g − (r) =[g1 − (r),...,g− n⎡(r)]P 1 +P + ⎢(t) ⎤⎥(t) = ⎣ . ⎦ ,P n + (t)⎡T1 −T − ⎢(t) ⎤⎥(t) = ⎣ . ⎦ .Tn −(t) The temperature rises⎡T 1 +T + ⎢(t) ⎤⎥(t) = ⎣ . ⎦ .T n + (t)at n ports of C ± and the powers⎡P − (t) =⎢⎣P − 1 (t)..P − n (t)⎤⎥⎦ .at the other n ports of C ± are defined by∫T + (t) = g +T (r)x + (r,t) dr,∫ΩP − (t) = g −T (r)x − (r,t) dr.ΩThe relation between port powers and temperatures is modelledbyan2n× 2n matrix h(t) and by its Laplace transform,the 2n × 2n matrix[ ]H11 (s) HH(s) =12 (s).H 21 (s) H 22 (s)Such definition of thermal network preserves both thepassivity and reciprocity properties 2, 3. In factProperty 5 (Passivity): For each time interval t 1 ≤ t ≤ t 2for C ± it results inW ± (t 2 ) ≤ W ± (t 1 )++∫ t2t 1(T +T (t)P +T (t)+T −T (t)P −T (t))dtEquivalently, the matrix H(s) is positive real.Property 6 (Reciprocity): In the Laplace transform domains, forC ± 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 inT +T1 (s)P + 2 (s)+T−T 1 (s)P − 2 (s) == T +T2 (s)P + 1 (s)+T −T2 (s)P − 1 (s)Equivalently, it is H 11 (s) =H T 11(s), H 22 (s) =H T 22(s) andH 12 (s) =−H T 21 (s).The following relation can be extablished between thethermal networks C and C ± .LetitbeThen it results ing + (r) =g − (r) = g(r) √2,P + (t) =T − (t) =P(t).x + (r,t)+x − (r,t)= √ 2 x(r,t)T(t) =T + (t)+P − (t).These relations show that the C thermal network can besynthesized by terminating C ± with a lossless multiport G,composed by means of gyrators as shown in Fig. 1. Also therelation between Z(s) and H(s) isZ(s) =H 11 (s)+H 22 (s)+H 12 (s)+H 21 (s)©EDA Publishing/THERMINIC 2008 54ISBN: 978-2-35500-008-9