A Thermal Model of a Forced-Cooled Heat Sink for Transient Temperature Calculations Employing a Circuit Simulator

Transcription

1 A Therml Model of Forced-Cooled Het Sink for Trnsient Temperture Clcultions Employing Circuit Simultor Uwe DROFENIK * Johnn W. KOLAR * Astrct. Power semiconductors cn e modeled s therml network of resistors nd cpcitors. The therml oundry condition of such model is typiclly defined s the het sink surfce temperture which is ssumed to e constnt. In relity, the het sink surfce temperture underneth the power module is not exctly known. In this pper we show how to set up therml model of the het sink in form of RC therml equivlent network tht cn e directly emedded in ny circuit simultor. The proposed therml het sink model tkes into ccount convection cooling, therml hotspots on the het sink se plte, therml time constnts of the het sink, nd therml coupling etween different power modules mounted onto the het sink. Experimentl results re given nd show high ccurcy of the het sink model with temperture errors elow %. Keywords: het sink, dynmic therml model, therml coupling, therml hot spot, het trnsfer coefficient Introduction. Therml Simultions Employing Circuit Simultor In order to optimie system design concerning incresing power density nd reliility issues, there is need to e le to perform esides numericl circuit simultion lso sttionry nd coupled trnsient numericl therml simultions. Generlly, power module nd its internl semiconductors cn e set up in good pproximtion s therml network consisting of therml resistors nd cpcitors. Such therml models cn e directly uilt into ny circuit simultor with minimum effort. The circuit simultor estimtes the semiconductor losses, nd the time ehvior of the losses is coupled with the therml model resulting in the time ehvior of the junction temperture ([], []. The therml oundry condition of such therml semiconductor model is typiclly defined s the het sink surfce temperture which is ssumed to e constnt. While lot of work hs een performed concerning the therml modeling of the power semiconductor (nd/or the power module, het sink models to e employed in circuit simultors re not common in power electronics, lthough the temperture-drop from het sink to mient might esily e in the rnge of the junction-cse temperture drop.. Defining Therml Model of the Het Sink Setting up simple therml model of het sink suitle for emedding it in circuit simultion considering therml hotspots therml coupling etween neighoring power modules dynmic ehvior (time constnts of the het sink convection cooling is difficult ecuse of the complex fin geometry, the threedimensionl temperture distriution, the impct of the fn chrcteristics nd the often complex nd difficult-to-model environment of the het sink within system environment. Furthermore, the trnsient therml impednce (nd/or therml resistnce of the het sink s experienced from the viewpoint of power module, is strongly dependent on the sie nd loction of this power module mounted onto the het sink. In this pper we propose method for setting up het sink model considering ll effects listed ove. The procedure works s follows: Tke het sink plus fn nd mount rectngulr test het source onto the center of the het sink se plte. Het up the configurtion nd mesure the sttionry temperture t se plte point close to the test source. Use geometry, mteril prmeters, nd the mesured temperture to prmeterie the equtions s given. Descrie the loction nd sie of the power modules to e plced on the het sink for the finl system design. Employ nlyticl equtions nd numericl finitedifference clcultions (no CFD needed! s descried. Get RC therml equivlent circuit of the het sink to e employing in circuit simultor. Bsed on very simple sttionry temperture mesurement n esy-to-use het sink model cn e derived. The necessry clcultions include trnsient numericl simultion of the temperture distriution inside 3Drectngulr lock of homogenous mteril which cn e done with FEM progrms employing only the het conduction eqution, ut lso with quite simple self-written finite difference code (FDM. Compred to otherwise necessry simultions of the het sink including the ir-flow (computtionl fluid dynmics - CFD, simultion times on tody s (4/5 PCs re reduced from few hours to less thn one minute. Furthermore, CFD simultions of het sinks with lrge numer of fins tend to e numericlly unstle nd often show wek convergence, while the FEM-simultions s employed for the therml models introduced in this pper show excellent numericl stility. First, we hve to find the het trnsfer coefficient of the ircooled het sink sed on se plte surfce temperture mesurement (section. This het trnsfer coefficient is essentil to set up simplified therml model of the het sink. In section 3, the simplified therml model will e employed to numericlly clculte therml step responses. This will e compred to two experimentl setups. In section 4, RC therml equivlent circuit of the het sink will e extrcted from the clculted step responses.

2 Het Trnsfer Coefficient of n Air-Cooled Het Sink. Finding the Het Trnsfer Coefficient of Het Sink The het sink temperture is defined y convective cooling which cn e generlly descried y het trnsfer coefficient h [W/m K] defined ccording to Q = A h T ( with the therml power Q [W], the totl surfce re (minly provided y the fins exposed to convection cooling A [m ], nd the temperture drop from fin surfce to mient. In cse of forced convection (which is the focus of this pper the het trnsfer coefficient h is strongly dependent on fn chrcteristic nd ir flow inside the cinet of the power electronic system. The proposed modeling procedure is sed on the ssumption tht the het flow from the fins into the ir cn e descried in good pproximtion y constnt het trnsfer coefficient h=const. ( ( Fig.: ( A CFD simultion shows the temperture field nd ir flow for het sink. In the vicinity of the power module there is hot spot. ( The simplified het sink model consists of plte with het trnsfer coefficient h=const s oundry condition t the ottom side, nd therml isoltion (h= t ll other surfces. As shown in [3], the three-dimensionl temperture field T(x,y, of plte with rectngulr het source locted t the center (Fig. together with Neumnn oundry condition (chrcteried y het trnsfer coefficient h_=_const t the ottom side _=_t nd therml isoltion (h_=_ t ll other surfces, cn y descried (y nlyticlly solving the three-dimensionl het conduction differentil eqution vi Fourier series s T ( x, y,, = T l= l= m= ψ (, cos( l 4 ψ (, cos( lm ψ ( x x m= cos( ψ (, cos( m y y ( with the coefficients 4Q ψ (, = ( t k h (3 k 8Q ψ l (, = sin π l k x ψ m cosh sinh l π ( x x l π ( x x ( cos( l π ( t h l π ( t ( ( ( l π k sinh l π t h l π t ( ( cosh( l π k 8Q (, = sin π m k y cosh sinh sin ( t h ( t ( ( ( k sinh t h t ( ( cosh( k 6Q ψ lm (, = sin π l m k x y ( y y ( y y ( cos( M = cosh ( t k h N = cosht N k h ( ( ( y y ( y y ( cos( l π ( x x l π ( x x ( cos( M N N (4 (5 (6 ( ( sinh ( t ( ( ( ( ( ( sinh t ( ( ( ( (7 (8 = (9 T PN - T [ C] C h = 54 W/m K x/ =.4 Q = 5W h [W/m K] Fig.: Dependency of the temperture t surfce point P N on the het trnsfer coefficient h [W/m K] for n ir-cooled het sink (=mm, =mm, t=.5mm with power module of x =4mm ( x/=.4, y =34mm nd Q=5W. The curve is derived nlyticlly from ( (9 nd/or (. He therml conductivity of the extruded luminium het sink is k = 5 W/mK. The temperture T=T PN T =.78 C is derived from sttionry CFD-simultion (y ICEPAK of the het sink shown in Fig.. Equtions ( (9 cn e esily implemented in ny progrmming lnguge, nd the numer of coefficients is dependent on the geometry rtios x/ nd/or y/. The

3 smller, e.g., x compred to, the more Fourier coefficients re necessry to descrie the power module geometry ccurtely. For detils see [3]. If the temperture t point t the het sink surfce, e.g. P N in Fig., is know, the het trnsfer coefficient h of the ircooled het sink cn e clculted from ( which is directly derived from (. T PN = T ( x = x, y = /, =, = T ( ( This cn e done, for exmple, grphiclly for certin power module ( x/ =.4 s shown in Fig.. First, sed on ( - (9 nd/or ( the temperture t certin point is plotted dependent on vrying het trnsfer coefficient h. In Fig. nd/or ( this is done for point P N close to the power module s shown in Fig., where temperture sensor cn e esily plced. In this exmple, the temperture t point P N is derived vi sttionry CFD simultion of the het sink shown in Fig.. With T PN =5.78 C t mient temperture T = C, the het trnsfer coefficient e found for this certin het sink plus fn configurtion s h = 54W/m K. The het trnsfer coefficient h derived this wy is dependent on the het sink fin geometry, the fn chrcteristic nd the ir-flow. It is not dependent on the power module nd, therefore, chrcteries the cooling of the het sink in very generl wy. The simplified therml het sink model with h=const t the ottom surfce s employed here, does not tke into ccount the irflow direction which distorts the temperture field (see Fig.(. In spite of these shortcomings, employing constnt vlue of h is justified for mny different het sink types s shown in the following sections.. Prmeter-Sensitivity Dependent on the Point of Sttionry Temperture Mesurement If the proposed method is to e employed in prcticl design, it is essentil to mke sure tht the mthemticl method to derive the verge het trnsfer coefficient shows roustness ginst mesurement inccurcies. Figure 3 shows the grphicl method s demonstrted in Fig. for different test het source geometries nd for different points of mesurement t the het sink surfce. The het trnsfer coefficient h is not dependent on the het source geometry. 8 ( x/ =.49 Q=5W x=48mm, y=68mm h = 54 W/mK P K P N P 6 8 h [W/m K] ( (c x/ =.4 Q=5W x=4mm, y=34mm h = 54 W/mK P K P N P 6 8 h [W/m K] x/ =.43 Q=5W x=4.8mm, y=6.8mm h = 54 W/mK P K P P N 6 8 h [W/m K] Fig.3: Centered test het sources of different sie (chrcteried y x/-rtio result in different temperture mesurements t selected points P, P N, P K on the het sink with =mm, =mm, t=mm, k=5w/mk (see Fig.(. The het trnsfer coefficient h descries only the convection cooling of the het sink vi the fins nd is, therefore, not ffected y the test het source geometry. Generlly, since the temperture distriution T(x,y, is proportionl to the power P V s cn e directly seen from ( (9, the ccurcy of the mesurement of h cn e incresed y simply incresing the therml power Q. Thermocouples tht re typiclly ville in power electronics lortory show solute errors in the rnge of ±.5 C. Prcticl limits of incresing the heting power re set y the mximum tempertures of the employed mesurement equipment. The center of the test het source (P shows the mximum temperture of the whole experimentl rrngement which is difficult to mesure. A hole hs to e drilled into the het sink se plte directly elow the test het source to insert the thermocouple. Alterntively, temperture sensor must e integrted into the test het source. Both methods chnge the temperture field, distort the temperture mesurement nd result in n incresed temperture mesurement error s discussed in detil in section 3. The proposed method offers the significnt dvntge to mesure the sttionry temperture t ny point of the het sink se plte. Therefore, mesuring the temperture close to the test het source (point P N in Fig. will give n solute temperture close to the mximum temperture occurring t the center of the test het source, ut will e esy nd ccurte to mesure y simply pressing the

4 thermocouple t point P N ginst the het sink surfce. For lrger test het sources (lrger x/-rtios, see Fig.3( the temperture t P N is much closer to the mximum center point temperture t P s compred to very smll test het sources (Fig.3(c. This mkes lrge test het sources generlly more ttrctive for this kind of mesurement. dt/dh [(Km /W] Q=W Q=5W Q=5W Q=W h [W/mK] Fig.4: The derivtive dt/dh of the het sink surfce temperture dependent on the het trnsfer coefficient h nd the heting power Q [W]. These curves re independent from x/-rtios of the test het source nd lso independent from the het sink surfce point of the sttionry mesurement. The curve of dt/dh for Q=5W shown here is vlid for ll curves of Fig.3. Concerning the ccurcy of the vlue of h, dt/dh of the curves in Fig. nd/or Fig.3 should e s lrge s possile. As shown in Fig.4, the derivtive is independent from the sie of the test het source nd the point of temperture mesurement, nd proportionl to the heting power Q [W]. Figure 4 is sed on the nlyticl model of the het sink s descried y ( (9. With rel het sink, setting h constnt is n pproximtion tht sometimes does not work well t points P K or P K (Fig. t the edge of the se plte (see lso section 3. It is, therefore, lso under this spect preferle to mesure the temperture close to the het sink t point P N. According to Fig.4, for the given het sink nd heting power Q=5W the vlue of h=54w/m K results in dt/dh -.6(m K /W. This mens tht with n solute temperture mesurement error of, e.g., T=±.5 t ny se plte point, the vlue of the clculted het trnsfer coefficient h for the het sink model will vry y out h= ±3W/m K. Douling the heting power Q will increse dt/dh to -.3(m K /W nd reduce the error of the het trnsfer coefficient h ccordingly y fctor of. 3 Clculting Therml Step Responses Bsed on the Proposed Het Sink Model 3. Exmple I: Hollow-Fin Cooling Aggregte The proposed procedure will e experimentlly tested employing hollow-fin cooling ggregte [4] s shown in Fig.5. Assuming tht the sie nd loction of the power modules of the finl system design is not known yet, simplified het sink model hs to e set up first s descried in detil in section. A test het source (W-resistor on 8mm copper het spreder is mounted onto the center of the het sink. After heting up nd reching stedy stte, the temperture on the het sink surfce close to the copper lock (e.g., point P N in Fig.( is mesured. Since the fn is in full opertion, the mesurement descries the forced convection ir cooling s it will e employed in the finl system design. If the operting environment of the het sink in the finl system (e.g., distorted ir flow inside the housing is lredy known, the ccurcy of the whole modeling scheme cn e incresed y performing the mesurement in comprle environment. Fig.5: Hollow-fin cooling ggregte (5x8x8mm 3,.5mm se plte thickness, luminium with k=5w/mk with fn. A test het source of Q=W is mounted onto the center. The shown wire is connected to thermocouple inserted in the copperhet spreder elow the heting resistor to mesure the center point temperture (P in Fig.. The heting resistor is not connected to voltge source yet. One sttionry temperture mesurement t just one se plte surfce point is sufficient to clculte the het trnsfer coefficient h employing the procedure descried in section. For testing purpose, the temperture ws mesured t four different points P, P N, P N, P N3, P K nd P K s shown in Fig.6. Employing ( nd/or Fig. we get the vlues of h s given in T.. The mient temperture is T =4 C nd the heting power is Q=W. point coordinte x [mm] coordinte y [mm] mesured h [W/m K] from ( P 6 P N P N P N P K P K T.: Coordintes (x is in ir flow direction, mesured tempertures nd resulting het trnsfer coefficients for different points t the het sink surfce. Further prmeters re t=mm, =5mm, =8mm, x=5mm, y=53mm, k=5w/mk. Note tht the coordintes given here ccording to the coordinte system in Fig.6 re different from the coordinte system of Fig.( which hs to e employed if working with equtions ( (9. Idelly, ll vlues of h should e equl. The simplified het sink model (Fig.( does not tke into ccount ir flow direction. Since the ir is heting up long the fins, the het sink temperture must generlly rise long the x-direction (ir flow direction in this exmple. This is why the mesured temperture t P N3 is higher thn temperture t P N or P N. Accordingly, the het sink coefficient clculted from P N3 -mesurement must e lower. The sme is true

5 for P K nd P K. The mesurement t the center point P hs een performed with thermocouple inserted into hole drilled into the copper het spreder of the test het source. While temperture mesurements t ll other points re performed y simply pressing the thermocouple onto the se plte surfce, performing P -mesurement provides dditionl therml resistnces of the copper lock nd of the therml grese etween het sink nd copper lock (λ.w/mk, thickness d 3µm. This dditionl therml resistnce increses the mesured temperture t P y out 4.5 C resulting in n inccurtely reduced vlue of h. The properties of the therml grese where derived y compring the sttionry experimentl mesurement to FEM simultion nd re in good ccordnce with vlues typiclly given in dtsheets. To set up the simplified therml model of the hollow-fin cooling ggregte, the verge vlue of h from the points P N, P N, nd P N3 is formed s pproximtely W h 65 m K = ( The simplified therml model of the hollow-fin cooling ggregte consists of n luminium lock with the het sources mounted onto it s shown in Fig.6. Now loction, sie nd numer of the power modules of the plnned system design hve to e defined in order to proceed with the modeling. The thickness of this lock is not equl to the se plte of the het sink ut hs to tke into ccount the fins. The fins typiclly provide significnt mss tht cts s therml cpcitnce nd, therefore, hve strong influence on the therml time constnts of the het sink. Furthermore, efore the het cn flow from the het sink into the cooling ir, the het hs to flow prtly through the fins, which increses the therml resistnce of the het sink. The fins lso increse the therml coupling of two het sources mounted onto the het sink in cses where the het sources re mounted ove the sme fins. Therefore, the fins hve to e considered in form of n increse of the thickness t of the simplified model. The se plte mss of the hollow-fin cooling ggregte is 3 kg mbp = ( m 8 3 =. 353kg ( Since the totl mss of the het sink ws mesured s.6kg, the thickness of the simplified model in Fig.6 hs to e y fctor of 3.4 higher thn the se plte thickness resulting in t = mm (3 Bsed on the simplified therml het sink model of Fig.6, the therml step responses of the vrious power modules locted there hve to e found. This cn most effectively e done y trnsient numericl temperture field simultion. We currently employ commercil 3D-FEM softwre (ICEPAK where we hve to solve only the het conduction eqution ecuse there re no fluids in the model of Fig.6. Insted of simulting the ir flow where the simultor hs to solve five differentil equtions (mss conservtion, energy conservtion, impulse conservtion in vector form simultneously, we now hve only one differentil eqution m to solve (het conduction eqution = energy conservtion. Also, the very complex meshing of the fins nd the chnnels etween the fins is voided. Therefore, the simultion time of the trnsient step response is reduced from more thn one hour for full scle CFD (computtionl fluid dynmics simultion to seconds for the simplified model shown in Fig.6. Wht is even more importnt is tht this very fst simultion shows excellent numericl convergence while the CFD simultion tends to e numericlly unstle nd/or gives inccurte results. Fig.6: Simplified therml het sink model of the hollow-fin cooling ggregte (Fig.5 ccording to Fig.( with h=65w/m K nd t=mm, =5mm, =8mm, k=5w/mk. In the FEM simultion the ottom wll is defined employing Neumnn oundry condition with h=65w/m K=const, ll other wlls re defined s thermlly isolting. The het sources re modeled s D-elements with continuous het distriution. Alterntively to employing FEM softwre, much esier to progrm finite difference methods (FDM will give ccurte results especilly due to the simple geometry of the simplified het sink model (only one homogenous lock with homogenous oundry conditions nd rectngulr D het sources. Since we work on utomting the modeling procedure descried in this pper, we will implement n ccording FDM code s it is well known from the literture ([5]. Writing CFD code for such project would increse the complexity of the softwre, the time effort nd the worklod on n unrelisticlly lrge scle. Fig.7: For testing the theory, three W-het sources re mounted onto the het sink. Ech het source consists of heting resistor on 5x53mm copper het spreder of 8mm thickness contining.5mm dimeter hole with n inserted thermocouple for temperture mesurement. The het sources re leled HS, HS, nd HS3 from the left to the right (opposite direction of the irflow, see lso Fig.6. The spce etween two neighor het sources is 5mm.

6 ( ( Q HS = W Q HS = W Q HS3 = W HS HS HS3 HS HS HS3 HS3 resistnce of copper lock nd therml grese. The therml step response of the het source tht is heting up (e.g., HS in Fig.8(, HS in Fig.8( nd HS3 in Fig.8(c is lwys distorted in the time rnge elow out one minute. This effect indictes n dditionl therml cpcitnce close to the ctive het source which comes from the heting resistor nd prtly from the copper het spreder (oth not covered y the simplified model of Fig.6. Employing flt het sources, e.g. power semiconductor chips, would hve resulted in more ccurte trnsient mesurements. This hs, however, no relevnce for setting up our simplified therml het sink model, ecuse this is sed on sttionry temperture mesurement directly on the het sink se plte surfce close to the test het source ut not inside the copper het spreder. The temperture errors of the simplified het sink model re elow % compred to the experimentl results in Fig.8 for the temperture rise of the single het source tht is eing heted up. The errors of the temperture increses of the other two het sources due to therml coupling re lrger (up to % ut the model lwys predicts higher tempertures from therml coupling effects which gurntees sfety mrgin in the therml design process. The reson for this lwys higher temperture prediction for therml coupling is tht the het flows not only through the se plte ut lso through the fins. In the proposed simplified therml model the fin mteril is employed to increse the thickness t of the model plte. In relity, fins hve n orienttion (prllel in ir flow direction nd conduct het only in one direction in n effective wy. This effect cn e considered in the simplified model (Fig.6 y mking the therml conductivity dependent on the direction. (c HS HS Fig.8: Therml step responses of ll three het sources HS, HS nd HS3 for heting ( only HS with Q=W, ( only HS with Q=W nd (c only HS3 with Q=W. The connected dots re experimentlly mesured, the solid lines re resulting from trnsient FEM simultions of the simplified het sink model. The dshed lines re resulting from the RC therml equivlent network model s descried in section 4 nd in the Appendix. In order to vlidte the procedure experimentlly the setup of Fig.6 is relied s shown in Fig.7 nd experimentl results re given in Fig.8 (connected dots. Results of the simulted (FEM therml step response from Fig.6 re shown in Fig.8 s solid lines. The mesured tempertures of the thermocouples hve een corrected ccording to the dditionl temperture drop cused y the therml 3. Exmple II: Extruded Het Sink As nother exmple, n extruded het sink (Fig.9 is tested experimentlly in nlogy to the previous section. Compred to the hollow-fin cooling ggregte, the ir flow is directed from the fn t the ottom side directly ginst fins nd se plte which results in more nonhomogenous cooling effect nd, therefore, lso in more non-homogenous het trnsfer coefficient. Furthermore, the se plte is thinner compred to its length nd/or width. In spite of this, the simplified model ssuming h=const gives ccurte results lso for this het sink s will e shown in the following. From sttionry temperture mesurements t the se plte close to the centered test het source (e.g. P N in Fig., we receive for the chrcteristic het trnsfer coefficient of this het sink (with prmeters t=.7mm, =5mm, =77mm, x=5mm, y=53mm, k=5w/mk W h m K = (4 to e employed s oundry condition in the simplified het sink model (Fig.. The se plte thickness of 5mm hs to e increse y fctor.54 to tke into ccount the mss of the fins resulting in t=.7mm for the simplified model. In Fig. test rrngement of there different het sources is set up to e tested ginst the results of the experimentl setup shown in Fig..

7 48 Q HS = 55W 3 HS HS 8 Fig.9: Extruded het sink (5x77x6.5mm 3, 5mm se plte thickness, luminium with k=5w/km with fn mounted onto the ottom side (only the wires of the fn re visile in the photo. A test het source of Q=W is mounted onto the center. The shown wire is connected to thermocouple inserted in the copperhet spreder elow the heting resistor to mesure the center point temperture (P in Fig.6. ( 4 48 Q HS = 75W HSc HS HS 3 HSc 8 4 Fig.: Simplified therml het sink model of the extruded het sink (Fig.9 with h=w/m K, t=.7mm, =5mm, =77mm, k=5w/mk. The D het sources re sied nd locted s descried in Fig.. ( 48 Q HSc = 55W HSc HS 3 8 HS 4 Fig.: For testing the theory, three different het sources re mounted onto the het sink. The het sources re leled HS, HS, nd HSc from the ck to the front (see Fig.. The center point coordintes of these three het sources re HS (-38.5mm/- 4mm, HS (/ nd HSc (57.5mm/47mm. Ech het source consists of heting resistor on copper het spreder of 8mm thickness contining.5mm dimeter hole with inserted thermocouple. HS nd HSc (oth emitting 55W therml power hve 33x5mm het spreder re, HSc (75W hs het spreder re of 53x5mm. Note tht the coordinte system employed here nd lso shown in Fig. is different to the coordinte system of Fig.( tht hs to e employed if ( (9 re used. The experimentl results of the therml step responses (connected dots in Fig. re in good greement with the results from the trnsient numericl simultion of the simplified het sink model (solid lines. (c Fig.: Therml step responses of ll three het sources HS, HS nd HSc for heting ( only HS with Q=55W, ( only HS with Q=75W nd (c only HSc with Q=55W. The dots re experimentlly mesured, nd the solid lines re resulting from the trnsient FEM simultion of the simplified het sink model. The dshed lines re resulting from the RC therml equivlent network model s descried in section 4 nd in the Appendix. 4 Therml Equivlent Circuit of the Het Sink Bsed on the Impednce Mtrix Model One wy to set up simple equivlent therml network model sed on the het conduction eqution is employing the impednce mtrix method [6]. The underlying mthemticl principle is superposition of different het sources ssuming liner differentil eqution. Strictly

8 speking, the het conduction eqution is not liner differentil eqution ecuse properties like therml conductivity nd therml cpcity re temperture dependent. Since this dependency is not very strong within temperture rnges s typiclly found in power electronic operting rnges, pplying superposition is justified in most cses. Ech het source hs to e heted up, nd the temperture rise (therml step response of this het source, ut lso of ll other het sources, hs to e mesured (see Fig.8 nd Fig.. In the following, we will write AB to indicte tht heting up het source B will hve n effect on the temperture of the het source locted t A s descried y the trnsient therml impednce AB. Since ech of n het sources mounted onto het sink influences the tempertures of ll other het sources, the totl numer of therml step responses to e recorded or clculted is n. The scheme cn e descried y mtrix eqution s T T T HS HS HS 3 = Q Q Q 3 (5 in cse of the hollow-fin cooling ggregte of section 3.. Here, the therml impednce is the normlied (divided through the therml power Q HS therml step response of HS in Fig.8(, where HS is heted up with Q HS =W. The step response of HS in the sme figure would give fter normlition (dividing through Q HS the trnsient therml impednce, nd so on. Figure 3 shows how to implement the mtrix eqution in circuit simultor. Therml power emitted from the power modules is modeled s current provided y signl-controlled current sources Q HSi, nd the trnsient therml impednces ji re modeled s RC-circuits. The voltge t the input side of such RC-circuit represents the prtil temperture rise T ji cused y het source HSi. Due to the principl of superposition ll prtil tempertures T ji must e dded to form the temperture rise T HSj of the module cse (t its center compred to mient. Q HS Q HS Q HS3 3 T T T 3 T HS Fig.3: Generl scheme of the RC therml equivlent network of het sink with three power modules mounted onto the se plte. In this figure only the network representing the temperture formtion of power module HS is shown. Ech current source represents the totl losses (therml power of one power module. The RC-circuits in the oxes re modeled ccording to the therml step responses derived vi trnsient FEM simultion of the simplified het sink model. In the Appendix implementtions of ji for oth experimentl het sinks of section 3 re given. The impednce mtrix grows with the squre of the numer of power modules. In this exmple there re 9 mtrix entries for just three power modules. For lrger numer of power modules on one het sink, the numer of necessry RC-representtions modeling the mtrix entries grows quickly nd will incresingly slow down the circuit simultion. It is, therefore, essentil to keep the numer of single RC-cells of ech mtrix entry s low s possile. There re widely used nd well known procedures to extrct RC-equivlent circuits from mesured or simulted therml step responses. These methods re highly ccurte ut result in lrge numer (typiclly 4 - of single RCcells. In this pper we employed serch lgorithm in order to find the optimum prmeter set (R- nd C-vlues to fit the reference step response (FEM-simultion from the simplified het sink model, solid lines in Fig.8 nd Fig. with minimum error for given structure nd cell numer. For given three-cell Cuer circuit, the serch lgorithm found the prmeter vlues s given in Fig.4 for the trnsient therml impednce from section 3.. Network structures nd prmeter vlues for ll 8 therml step responses clculted nd mesured in section 3 re given in the Appendix. Therml step responses of these RC-networks re shown in Fig.8 nd Fig. s dshed lines nd re in very good greement with the FEM-simultion (solid lines. Incresing the RC-cell numer of the single mtrix entries would eliminte the very smll remining inccurcies ut this would not mke much sense ecuse of the generl inccurcies of the simplified het sink model in the rnge of 5 %. The representtion of Fig.4 is sed on mthemticl curve fitting nd provides prtil temperture tht does not exist in relity. It is mthemticl model with no rel physicl mening (lthough it is Cuer-type equivlent circuit. Fig.4: Possile implementtion of the trnsient therml impednce in circuit simultor. With the current t the input side representing the power Q HS emitted y het source HS, the voltge drop from input side to ground represents the prtil temperture rise (ginst mient T = Q HS. For symmetry resons there is lwys AB = BA which reduces the numer of different mtrix entries. In cse of the hollow-fin cooling ggregte there is n dditionl geometric symmetry etween HS nd HS3 tht further reduces the numer of different mtrix entries. The network shown in Fig.3 clcultes temperture differences T HSi from the het sink elow the power module HSi to mient temperture T. The tempertures T T HSi represent the het sink temperture (relied in the circuit simultion in form of voltge-controlled voltgesources for the therml model of the power semiconductor tht is independently modeled in the circuit simultor. Agin, the underlying principle is superposition nd one cn directly numericlly clculte the power semiconductor

9 junction tempertures under considertion of the therml ehvior of the het sink. Generlly, the therml models of semiconductor (including therml grese nd het sink hve to e coupled vi signl-controlled current- nd voltge sources, ut must not e coupled directly when pplying the impednce mtrix. Conclusion The pper proposes generl RC therml equivlent network model of het sink to e esily emedded in ny circuit simultor. The network model considers convection cooling, therml hotspots elow the power modules, therml time constnts introduced y the het sink, nd therml coupling etween different power modules mounted onto the se plte. Experiments for two different het sinks show temperture errors elow %. The proposed procedure is complex ut cn esily e utomted in form of softwre. Currently such softwre tool is under development t the Power Electronic Systems Lortory, ETH Zurich. The input to this pckge is the het sink geometry nd one sttionry temperture mesurement. The output will e the therml RC equivlent circuit redy for emedding in ny circuit simultion. The whole computtionl effort of the proposed modeling procedure should e in the rnge of just few minutes. References [] U. Drofenik nd J. W. Kolr, Therml Anlysis of Multi-Chip Si/SiC-Power Module for Relition of Bridge Leg of kw Vienn Rectifier, in Proceedings of the 5th IEEE Interntionl Telecommunictions Energy Conference, Yokohm, Jpn, pp , Oct. 9-3, 3. [] U. Drofenik nd J. W. Kolr, A Generl Scheme for Clculting Switching- nd Conduction-Losses of Power Semiconductors in Numericl Circuit Simultions of Power Electronic Systems, Proc. of the 5 th Interntionl Power Electronics Conference (IPEC, Niigt, Jpn, April 4 8, 5. [3] G. N. Ellison, Mximum Therml Spreding Resistnce for Rectngulr Sources nd Pltes With Nonunity Aspect Rtios, in IEEE Trnsctions on Components nd Pckging Technologies, Vol. 6, No., pp , June 3. [4] Fischer Elektronik GmH, Extruded hetsinks: SK, dtsheet pulished t [5] S. V. Ptnkr, Numericl Het Trnsfer nd Fluid Flow, ISBN , Tylor & Frncis, 98. [6] T. Frnke, G. Ziser, J. Otto, M. Honserg-Riedl, R. Sommer, Current nd Temeprture Distriution in Multi-Chip Modules under Inverter Opertion, in Proceedings of the 8th Europen Conference on Power Electronics nd Applictions (EPE, Lusnne, Switerlnd, 999. Z _ = Z 33_ Z _ = Z _ = Z 3_ = Z 3_ Z 3_ = Z 3_ Z _ = Z _ = Z 3_ = Z 3_ Z _ Z 3_ = Z 3_ = Z _ = Z _ Z 3_ = Z 3_ Z 3_ = Z 3_ = Z _ = Z _ Z 33_ = Z _ T.: Possile entries of the impednce mtrix of (5 representing the hollow-fin cooling ggregte (section 3.. The vlues re found y serch lgorithm, the therml step responses re shown in Fig.8 (dshed lines. They re in very good greement with the reference curves from the FEM-simultion of the simplified therml het sink (solid lines in Fig.8. Z _ Z _ = Z _ Z 3_ = Z 3_ Z _ = Z _ Z _ Z 3_ = Z 3_ Z 3_ = Z 3_ Z 3_ = Z 3_ Z 33_ T.3: Possile entries of the impednce mtrix of (5 representing the extruded het sink (section 3.. The vlues re found y serch lgorithm, the therml step responses re shown in Fig. (dshed lines. They re in very good greement with the reference curves from the FEM-simultion of the simplified therml het sink (solid lines in Fig..