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transistor, through the bulk and beyond. This observationrises a relevant question concerning how important is theinfluence of the carrier in terms of thermal efficiency andwhat is its impact on the transistor functioning. This issue isaddressed in section VI.IV. FROM 3D FINITE ELEMENTS THERMAL SIMULATION TOREDUCED SYSTEMThe first order numerical system resulting from the heattransfer, the various boundary conditions (heat sources,baseplate temperature) and the FE method is given by:C . X & = −K.X + Fu(2)T = E.X(3)where C is the heat capacity matrix, K the heatconductivity matrix, F the load vector, u the Heaviside stepfunction and X the vector of temperature nodes.Because the size of the numerical system is large, thechoice of the output nodes to determine the equivalentoperating temperature T is important. This choice is madethrough the selection matrix E.The global system is not solved in ANSYS directly.Instead, Mor4Ansys [9] is used to extract the matrix systemfrom ANSYS binary files. Then, a Ritz-Arnoldi model orderreduction (MOR) technique using projection [11] [12] isapplied to obtain a reduced systemC . X K X F uR& = − . +R R R R(4)TT = E . XR R(5)The ‘ R ’ subscript indicates a reduced matrix. The newsystem takes the same form as the original system but with amatrix dimension significantly reduced. This reduced systemis the starting point to define the thermal impedance. Fig. 6compares the results from ANSYS simulation and thosegiven by the reduced thermal model using Ritz-ArnoldiMOR. Only five temperature nodes have been retained forthis reduction (matrix E). They correspond to the maximumtemperature reached in each bi-cell.Temperature (°C)Fig. 6ANSYS simulation for temperature distribution within the deviceand result given by the reduced thermal model for a GaAs HBTbrazed on a copper carrier, T 0=20°CV. THERMAL IMPEDANCE DETERMINATIONTo proceed to Z th determination, the temperature T R (5) isrepresented in the Fourier domain:T− 1T = E .( jϖ. C + K ) . FRR R R(6)If we consider a unique temperature for representing heat24-26 September 2008, Rome, Italyelevation within the transistor, Z th can be easily formulatedby:Temperature (°C)Z = (7)th PP is the dissipated power within the transistor. Arbitraryand in the most case, T R is taken as the maximumtemperature reached in the transistor. This choice is logicallyjustified because it represents the worst case.However, by only considering the maximum temperature,the thermal behavior of a multi-fingers transistor is onlylocally described. In multi-finger devices, the electricalbehavior of each finger is not impact the same way preciselybecause of the non homogeneous thermal distribution. Ascan be seen on Fig. 6 the temperature between the extremefinger and the central one can rise up to 10°C for a dissipatedpower of 1.5W which corresponds to the dissipated power inthe application concerned for the transistor. Like theelectrical model which has to describe the whole transistor atits access, the thermal model has also to take into account ofthe contribution and the coupling phenomena betweenfingers for representing the transistor thermal behavior.It is why in our case, a temperature averaging is used. Thenodes retained for the average temperature are the meshnodes used for the emitter, the base and the collectorvolumes. Z th is extracted by the quotient of the averagetemperature divided by the total input power.Zth=1NRT R×PFig. 7 compares the results given for the thermalimpedance by considering either the maximum temperatureTmax (the first hypothesis) or the average temperature for allthe finger nodes (N) (the second hypothesis). We canobserve that the reality, i.e the measurement, is inside thesetwo boundaries. So we proposed another hypothesisconsidering an average temperature for the selected nodes(S). Results are shown in Fig. 8. This selection of nodes Sconsists in avoiding side effects by removing the external bicelland the extremities of others bi-cells as shown in Fig. 9.Re{Zth} and Im{Zth} (°C/W)35302520151050-5-101 10 100 1000 10000 100000Fig. 7N R∑i=1Comparison between the maximum temperature (Tmax) and thetemperature average of the nodes (N) of all fingers used for thethermal impedance determinationTR(8)©EDA Publishing/THERMINIC 2008 192ISBN: 978-2-35500-008-9