Online proceedings - EDA Publishing Association

Dissipatedpower P(t)R1 R2 R3 R4 R5C1 C2 C3 C4 C524-26 September 2008, Rome, ItalyDissipatedpower P(t)Fig. 1: Foster type thermal one-port network (2 terminal device) servesas connecting impedances in model of Fig.4 and Fig.7.which include the generalised sources introduced in (1): s n =( p l , T a,c , J k ), which may also depend nonlinearly on the localtemperature T n (t), the result of (1) can be presented by asimplified thermal equivalent circuit. A similar thermalequivalent circuit has been presented in [12] using a verydifferent derivation by starting from discretized FEM or finitedifference equations of the set-up and making use of themultipoint moment matching reduction method. The simplifiedmodel/circuit has the disadvantage, that it is not straightforwardto assemble complex systems from subsystems. Inorder to combine thermal models representing subsystems itis preferable to have models which allow for the determinationof both: the temperatures at specified locations or thermalcontacts and the associated heat flows at the contact areas.A model that fulfils these criteria and is rigorous in thesense discussed in the introduction has been introduced in[6] and extended to the dynamic case in [9]. The model isdescribed by a network represented in Fig.2 for the specialcase of three thermal contact areas. The general constructionprinciples for the network are: Each thermal contact area isrepresented by one node. Then one junction node is introduced(where the temperature is to be monitored) and oneextra node to be connected with the heat current source P.All nodes (contacts, junction, heat source) are connected directlyby thermal resistors in the case of the steady statemodel. As has been shown in [6] the connection of the heatsource to the extra node P instead to the junction node makesthe network exact. When m denotes the number of thermalcontacts, the network obtained in this way has (m +2) (m +1)/2 thermal resistors, i.e. two parameters more than the compactmodel of [6]. The linear fit of the model parameters tomeasurement or simulation data and the subsequent determinationof the resistor values by analytical relations is easyand fast [6]. The resistor values are not all uniquely determinedby the model parameters, since there are two resistorsmore than independent model parameters.For the dynamic thermal model the same network was established[9] with the resistors being replaced by Cauer ladders,shown in Fig.3. The Cauer ladders form three terminalnetworks with one node grounded to an external referencenode. It is thus achieved that the heat flow(dissipated power)PTjuncT 1 T 3J 1 T 2J 3J 2Fig. 2: Thermal network for compact model with 3 thermal contacts.Each straight line connecting nodes represents one thermal resistor incase of steady state model or one Cauer sub-circuit, Fig.3, in case oftransient model.Fig. 3: Cauer type thermal network with 3 thermal contacts for transientcompact model in Fig.2. One terminal is grounded to the temperaturereference node (ambient).entering the device at node P does not necessarily leave atthe same time at the contact terminals during heating up orcooling down of the device. The missing current (heat flow)rushes through the grounded terminal of the Cauer ladders tosatisfy current conservation. It is difficult to adjust the parametersof the network of Fig.2, i.e. the thermal resistorsand capacitors of the Cauer ladders, to measured or simulateddata, because no analytic treatment or linear fit is possible.It is straight forward to extend the steady state and transientmodel of Fig.2 to several heat sources [6, 9] by addingnodes for the additional heat sources and connecting them inthe same way by direct links to the remaining nodes.III. IMPROVED MODELIn order to circumvent the difficulties with the Cauer laddersa new exact network is proposed and investigated in thispaper. For the transient case the Cauer ladders in Fig.2 arenow replaced by one-port impedances (two-terminal entities).A realistic thermal transient behavior, which allows forviolation of heat current conservation between heat inflowand outflow at the thermal contacts, is obtained by addingone node connected to the external reference node and whichdoes not correspond to a device thermal contact. As beforeall nodes are connected directly, now with thermal one-portimpedances. An example circuit is shown in Fig.4 with twothermal device contacts and one heat source.Networks built up from R, C, L elements are efficientlydescribed in Laplace domain by rational functions in s = i ω,since the impedances of R, C, L are given by R, 1/(Cs), Ls,respectively. An old result of network theory cited in [18,p.55] states, that the behaviour of every linear passive n-terminal circuit can be represented by a n-node network withdirect connection of the n nodes by two-terminal impedances(one-ports) described by rational functions in s. The transferimpedances of the connecting links are not necessarilyphysically realizable with positive R, C, L elements. However,for thermal model-building the R, C, L elements are notrequired to be positive. In order to determine the transfer impedancesby fit to measurement or simulation data, it is moreappropriate to characterize the transfer impedances by theirreciprocal functions, the admittances y i k (s) , whereJ i (s) = y i k (s) T k (s) (5)is the (inward) current leaving at node (terminal) i, when thetemperature at node k is T k and the temperature at all othernodes is zero (reference temperature). For this boundarycondition the terminal current J i rushes only through thenetwork impedance −1/y i k (s), since the current for the otherimpedances −1/y i l (s) with l ≠ k is zero because of zero voltage(temperature) difference. Thus the −1/y i k (s) can be iden-©EDA Publishing/THERMINIC 2008 71ISBN: 978-2-35500-008-9