REDUCING THE POSSIBILITY OF SUBJECTIVE ERROR IN THE DETERMINATION OF THE STRUCTURE-FUNCTION-BASED EFFECTIVE THERMAL CONDUCTIVITY OF BOARDS

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1 Nice, Côte d Azur, Frace, Septeber 2006 REDUCING THE POSSIBILITY OF SUBJECTIVE ERROR IN THE DETERMINATION OF THE STRUCTURE-FUNCTION-BASED EFFECTIVE THERMAL CONDUCTIVITY OF BOARDS Erő Kollár, Vladiír Székely BUTE, Departet of Electro Devices, H-52 Golda Gy. tér 3, Budapest, Hugary <kollar szekely@eet.be.hu> ABSTRACT The theral respose fuctio give to a uit-step dissipatio accurately characterizes the theral syste. Istead of the theral respose fuctio the so-called structure fuctio describig three-diesioal as the equivalet odel of oe-diesioal heat-spreadig, created fro the theral respose fuctio with the help of coplex atheatical procedures, is ofte used. Usig the structure fuctio the partial theral capacity ad partial heat resistace of certai eleets of the theral syste ca be idetified. If the geoetrical easureets of a theral syste of siple geoetry ad hoogeeous aterial (such as a hoogeeous rod or board, etc.) are kow, the coefficiet of theral coductivity of the aterial i questio ca be deteried fro two poits of the structure fuctio at 2-5 per cet of accuracy. I this paper a ethod is preseted which applies a wide rage/sectio istead of two poits of the cuulative structure fuctio to deterie the theral coefficiet, thus reducig the subjective error derivig fro the selectio of the two poits. The above ethod is preseted ad illustrated i siulated as well as easured theral trasiet resposes. R Σ ( ) = R i i= ad the cuulative theral capacitace is: C Σ ( ) = C i i=, (), (2) where R i ad C i deote the eleet values of the i-th stage of the Cauer-type odel etwork. Plottig the C Σ vs. R Σ values results i the cuulative structure fuctio (Figure 2).. INTRODUCTION The structure fuctios are graphical represetatios of the RC-odel (Cauer-etwork) of the theral syste (Figure ). Figure : The RC-odel of the theral syste. Figure 2: The cuulative structure fuctio. The cuulative structure fuctio characteristic of board-like aterials of hoogeeous isotropic coductivity (Figure 3). The theral resistace betwee the -th eleet of the odel etwork ad the heat source (drivig poit) is: TIMA Editios/THERMINIC page- ISBN:

2 Erő Kollár, Vladiír Székely Reducig the possibility of subjective error i the deteriatio of the structure-fuctio-based... Cth [Ws/K] T3Ster Master: cuulative structure fuctio(s) = W/K THE -FINDER ALGORITHM The -fider algorith deteries the theral coductivity coefficiet fro the itegrated structure fuctio of the board-like aterials. The algorith exaies how the steepess of the graph (curve) chages if the syetrical iediate eighborhood of the give poit i the itegrated structure fuctio is syetrically exteded e Figure 3: The cuulative structure fuctio of hoogeeous board. I order to deterie the coductivity value o the liear rage fro the itegrated structure fuctio []: The steps of the algorith are the followig: Coductivity values are deteried i each poit of the rage accordig to (4) Cth( + ) π Cth( ) ( R ) = l, th (4) w 4 R th( + ) Rth( ) w C Σ2 l C = 4π R R Σ, Σ2 Σ (3) where w is the thickess of the board, ad i the idexes of R th ad C th idicates the R th ad C th values plus (+) saples towards the higher values ad ius ( ) saples towards the saller values fro the poit exaied. ( = 20). where w is the thickess of the board ad the R ad C idexes idicate the values belogig to the selected poits. The ai aspect of the selectio of the two poits is that both should be withi the iddle, liear (labda easurig) rage ad should ot be idetical. Our siulatios carried out by the SuRed [2] siulatio progra showed that cosiderig the criteria of the above etioed ethod as a basis ad keepig i id the aspects of accuracy these criteria are ot eough. I the selectio of the two poits there always reais a certai degree of subjectivity, the subjectivity of the perso carryig out the evaluatio, which appears as a error whe deteriig the theral coductivity. Therefore we have elaborated a ethod to reduce subjective error, which applies a wide rage/sectio istead of two poits of the cuulative structure fuctio to deterie the theral coefficiet [3]. It is a additioal advatage of the ethod that it deteries the iflectio poit of the selected rage ore accurately, tha deteriig it fro the secod derivative accordig to the R th. Fro the coductivity values belogig to a certai poit of the rage (R th ) average(s) (5) ad δ (R th ) relative deviatio(s) (6) are deteried. ( R ) ) = ), (5) = ( R ) ( ) )) = ( ) δ th = (6) th Describe the δ (R th ) relative deviatios ad (R th ) average theral coductivity(ies) i relatio to R th theral resistace. We seek iial values i the δ (R th ) deviatio fuctios. Where the δ (R th ) fuctios together give the sallest iiu, this δ (R th ) value belogig to the R th poit gives the effective theral coductivity o (R th ) fuctio as well as the iflectio poit of the itegrated structure fuctio. Note: The ax = 20 ad values are experietal oes. The algorith does ot exclude the use of ay other ax ad values. 2 TIMA Editios/THERMINIC page- ISBN:

3 Erő Kollár, Vladiír Székely Reducig the possibility of subjective error i the deteriatio of the structure-fuctio-based EXAMPLE I. I this exaple we started fro the siulatio of the theral trasiet heat respose of the --thick hoogeeous copper board i vacuu. We created a itegrated structure fuctio fro the gaied theral respose fuctio usig the T3Ster-Master [4] progra. The progra deteries coductivity fro two poits of the itegrated structure fuctio selected at rado ad a board thickess give at rado. The calculated theral coductivity of the saple ay differ fro the 390 W/K value set as siulatio paraeter eve with.5 per cet, depedig o the two selected poits (Figure 4). Cth [Ws/K] T3Ster Master: cuulative structure fuctio(s) = W/K e Figure 4: The itegrated structure fuctio of the -thick ECU57 saple siulated i vacuu Let us eploy the -fider algorith for the calculated itegrated structure fuctio ad describe the δ (R th ) relative deviatios i relatio to the R th theral resistace. Relative deviatio 3,5% 3,0% 2,5% 2,0%,5%,0% 0,5% =20 =5 =0 =5 The sallest deviatios 0,0%,0,2,4,6,8 2,0 2,2 2,4 Figure 5: The δ (R th ) relative deviatio fuctios of the saple The sallest deviatios are gaied at R th =.73 W/K. Describig the (R th ) average theral coductivity values i relatio to R th heat resistace we ca read the values belogig to the above theral coductivity resistace. Figure 6 shows the fuctio ad Chart suarizes the values read. The sallest iius of the relative deviatio fuctios desigate accurately that R th poit where the deteriatio of the theral coductivity value is advisable. With the help of the sallest iiu poits foud i the relative deviatio fuctio of theral coductivity we are able to prevet the appearace of subjective error at the deteriatio of theral coductivity value derived fro two poits. Theral coductivity [W/K] W/K = =5 384 =0 382 = K/W 374,0,2,4,6,8 2,0 2,2 2,4 Figure 6: The (R th ) average theral coductivity fuctios of the saple I certai cases it ight be ecessary to deterie the iflectio poit of the cuulative structure fuctio. How-ever, the deteriatio fro the secod derivative accordig to the R th is ot always a preferable solutio. The double derivatio ay icrease the oise the fuctio so uch that we ca select the iflectio poit oly i a subjective aer. The -fider algorith sooths dow this oise ad provides the iflectio poit of the cuulative structure fuctio at the sallest iiu poit of the deviatio fuctios. M δ [%] [W/K] R th [K/W] Chart : Theral coductivity of the ECU-57 saple deteried by the -fider ethod (w = ) TIMA Editios/THERMINIC page- ISBN:

4 Erő Kollár, Vladiír Székely Reducig the possibility of subjective error i the deteriatio of the structure-fuctio-based EXAMPLE II, EFFECTIVE THERMAL CONDUCTIVITY OF PCBS Fro the aspect of easureet techology the easure of the effective theral coductivity of PCBs is greatly siplified by the fact that it ca be carried out i the air as well ad it is ot ecessary to use vacuu. I that case, however, the shutig effect of the air ust soehow be take ito cosideratio. I this experiet we siulated the theral trasiet respose of --thick boards havig theral coductivity betwee 2 ad 4 W/K i air. Due to the shutig effect of the air the theral coductivity values of the board siulated i the air deteried o the basis of the structure fuctio are, of course, higher tha the values adjusted as the paraeters of the siulatio. The shutig effect of the air chages the derivig theral coductivity oly a little, but the size of this error ca be easured i relatio to the subjective error appearig at the selectio of the two poits. I the experiets we eployed the followig cylider-syetrical structure (Figure 7). Fro the aspect of easureet techology it is advisable to use such a group of graphs which gives the theral coductivity of the saple (that is, easured i vacuu) fro the theral coductivity values easured i the air (Figure 9). Theral coductivity i siulator / deteried theral coductivity 00% 99% 98% 97% 96% 95% 94% 93% 92% 9% 90% w = radius = 0 radius = 20 radius = Theral coductivity deteried withi the labda easurig rage, siulated i the air [W/K] Figure 9: Theral coductivity adjustet curves of the saple for the easy easurig 5. CONCLUSIONS Figure 7: The board theral coductivity easureet setup Usig the -fider algorith we deteried the relative icrease of the theral coductivity values of the board ad plotted it i relatio to the paraeter of the siulatio burdeed with the shutig effect of the air (Figure 8). Relative icrease of the theral coductivity deteried withi the labda easurig rage, siulated i the air 0% 9% 8% 7% 6% 5% 4% 3% 2% % 0% y = 3,57E-0x -,47E+00 R 2 = 9,75E-0 y = 3,80E-0x -,75E+00 R 2 = 9,54E-0 radius = 30 radius = 20 radius = 0 w = y = 4,75E-0x -2,39E+00 R 2 = 9,70E Theral coductivity as the paraeter of the siulator, Labda [W/K] Figure 8: The air surroudig the board icreases theral coductivity We have created a algorith which reduces the subjective error appearig at the deteriatio of the local theral coductivity based o structure fuctio of board-like aterials i a way that it applies a wide rage istead of two poits of the cuulative structure fuctio to deterie the theral coefficiet. We have show two applicatios for the algorith. This algorith is advatageous whe the subjective error is coesurable to other errors, for exaple the shutig effect of the air at the deteriatio of the effective theral coductivity of boards. 6. ACKNOWLEDGEMENTS This work was partially supported by the PATENT IST Project of the EU, ad by the 2/08/NKFP- 200 INFOTERM Projects of the Hugaria Goveret. 7. REFERENCES [] V. Székely, M. Recz, S. Török, ad S. Ress: Calculatig effective board theral paraeters fro trasiet easureets, IEEE Trasactios o Copoets ad Packagig Techology, Vol. 24, NO.4, pp , Deceber 200. [2] TIMA Editios/THERMINIC page- ISBN:

5 Erő Kollár, Vladiír Székely Reducig the possibility of subjective error i the deteriatio of the structure-fuctio-based... [3] M. Recz, A. Poppe, E. Kollár, S. Ress, ad V. Székely: Icreasig the accuracy of structure fuctio based theral aterial paraeter easureets, IEEE Trasactios o Copoets ad Packagig Techologies, Vol. 28, ISS, pp. 5-57, March [4] TIMA Editios/THERMINIC page- ISBN: