Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyThe Minimal Set of Parameters for Exact DynamicThermal ModelsYork Christian Gerstenmaier* and Gerhard Wachutka***Siemens AG, Corporate Technology, D-81730 Muenchen, Germany, e-mail: yge@tep.ei.tum.de**Institute for Physics of Electrotechnology, Munich University of Technology, GermanyAbstract- A thermal model is presented that follows from theheat conduction equation and is exact under the conditions thatthe mass density, the specific heat and the thermal conductivityin the set-up do not depend on temperature and that the individualthermal contact areas of the models have uniform temperaturedistribution. The network models allow for the determinationof both: the transient temperatures at specified thermalcontacts and the associated heat flows at the contact areas.Compared to previous models, the network of the reduced compactmodel consists of one-port impedance links between externalterminals and one reference node, which is explicitly added.When m is the number of thermal device contact areas and pthe number of heat sources, the model is characterized by (m +p +1) (m + p ) / 2 one-port impedances with its associated R, C,L elements. A methodology is investigated for the determinationof the network parameters, which poses in many cases a highlyill conditioned problem, which may render the results useless.Alternative methods are suggested.I. INTRODUCTIONIn recent years many contributions have addressed theproblem of describing the thermal behavior of electronic andother systems by reduced models and networks [1]. Numerouscompact steady state (e.g. [2-6]) and dynamic (transient,e.g. [7-12]) thermal models have been established for a rapidcalculation of temperatures. The notion of “compact” thermalmodel usually implies boundary condition independence(BCI) [2, 3], i.e. the model is valid for all (or nearly all) reasonabletemperatures, heat flows and also heat transfer coefficientsapplied to the thermal contact areas. An advantage ofthe model presented in [13, 14] is its ease of parameter determinationby simple linear least square fit to measured orsimulated heating curves (thermal impedances). A rigoroussystematic approach has been presented in [6] for steadystate models and in [9] for transient models, as an exact consequenceof the linear heat conduction equation. The modelsare exact under the conditions that the mass density, the specificheat and the thermal conductivity in the set-up do notdepend on temperature and that the individual thermal contactareas of the models have uniform temperature distribution.Non-uniform temperature and heat flow conditions atthe contact areas have been investigated in [15]. The modelparameters of the steady state model [6] can be determinedby linear fit to measured or simulated data. For the transientnetwork of [9], which includes thermal capacitors, no parameterextraction method was provided. It is the purpose ofthis paper to develop this dynamic model further and to providea parameter extraction method for a modified equivalentnetwork.II. THERMAL MODELS AND NETWORKSThe model in [11, 12] characterizes the thermal set-up bya set of M ≤ 20 effective time constants t i which are logarithmicallydistributed, typically between Min(t i ) = 10 -4 s andMax(t i ) = 1000 s (depending on system and heat source size),and besides this are chosen freely. The model equation forthe temperature field T(x, t) reads:M L+C+Jt∑ ∑ il ∫0li=1 l=1( τ t) / tT(x,t)= M ( x)⋅−s ( τ ) e i dτ(1)where s l (t) denotes a heat source term which can be eitherthe dissipated power p l (t) of a chip l, l =1, .., L or an appliedaverage ambient temperature T a,c (t) at a thermal contact areac = 1, .., C or a thermal heat flux J k (t) at a thermal contact k= 1, .., J. The M i l (x) represent the model parameters e.g. fora chosen set of locations x j , usually the hot spots of the systemin the chip centres. The M i l (x) values are obtained bylinear least square fits to unit step responses of FEMsimulatedor measured T(x, t) for individual s l (t). Completetransient temperature fields can be calculated quickly asshown in [13, 14]. Model (1) contains the thermal impedancematrix z jl (t) in the time domain (impulse responses):(2)L+C+JtM−t/ tT(xij,t)= ∑ ∫ z jl(t −τ)sl( τ)dτ, z jl(t)il ( x j)e0=∑Ml=1i=1By use of the Laplace transformation L{ T(t)}= T(s)=∫ ∞ −st = T( t)e dt , s = i ω, model (1, 2) can be expressed as:0L+C+JM Mil( x j )T(x j,s)= ∑ z jl ( s)sl( s), z jl ( s)= ∑(3)( s + 1/ tl=1i=1 i )with impedances z jl (s) which can be represented for given j, lby Foster type thermal equivalent circuits as shown in Fig.1with elements C i = 1/ M i l (x j ), R i = t i M i l (x j ), so that R i C i =t i . Also negative pairs of R i , C i are allowed to occur, as longas the product (the time-constants) are positive. Thus theusual Foster type representation [16, 17] of the z jl (s) is obtained:M 1z jl( s)= ∑ =, s = − = −ii 1/ ti1/( RiC1i)(4)Ck( s −si)With the help of modified heat sourcesNP i j ( t,.. Tl( t)..)= ∑ n =Min(x j ) t1 isn(t,Tn( t))©EDA Publishing/THERMINIC 2008 70ISBN: 978-2-35500-008-9