Difficulties on the estimation of the thermal structure functionfrom noisy thermal impedance transients

M. Salleras', J. Palacin" 2, G. Carles', S. Marco''Sistemes d'Instrumentacio i Comunicacions, Departament d'Electronica, Universitat de Barcelona

Marti i Franques 1, 08028 Barcelona, SPAIN2Departament d'Infoimatica i Enginyeria Industial, Universitat de Lleida

Jaume II 69, 25001 Lleida, SPAINemail: santi ehubes, Phone: +34934029070, Fax: +34934021148

Abstract constant. The Foster network has no physical meaningbecause of the capacitances interconnecting nodes. Forthis reason the Cauer equivalent network is preferred toinsight in the thermal pathflow of electronic components. assign physical meaning to the RC values. To compute

In this work the sensitivity to noise of this technique isanalzed.Forhis urpoe, ahomoeneos slb of this equivalent network the Foster-Cauer transformationanalyzed. For this1pupose, homof is used [4]. The result is a Cauer network with n-stagesmaterial is considered, since for this case the exact time (fig.1) where each stage consists on a capacitance tied toconstant spectrum iS known. Results show that the '~constant ~ ~~ . spctuiskon.eut hwta h ground, and a resistance connected to the next stage node.thermal structure function is very sensitive to noise in the g

thermal transient. R Rn

1. Introduction 0

Thermal characterization of electronic components isan active research field. Microelectronic industry trends C, cpredict higher number of components in smaller areas. R} R 16This makes failure prediction and analysis more difficult. RMoreover with higher operating speeds and lowervoltages, devices become extremely sensitive to minor cOT ci2 C;;ydeffects. Most experts on the field claim for a real timeanalysis of defects through non-destructive techniques. Figure 1: Foster (top) and Cauer (bottom) networksSome techniques exist which accomplish these goals with n-stages.such as X-Ray microtomography or Scanning AcousticMicroscopy [1-3]. However, these techniques involve In this work the computation of the structure functionsophisticated equipment and a certain level of expertise will be discussed. In section 2, the ill posed Foster-Cauerwith the methodology. transformation will be explained and the structure

In the last decade thermal transient measurements function properties defined. A reference model to test thehave been used to evaluate die attach quality and structure function computation will be presented insoldering failure in electronic components through the so- section 3. Last, section 4 will present some resultscalled structure function proposed by Szekely [4]. The obtained concerning the structure function sensitivity tostructure function is directly computed from the thermal thermal transient noise.transient evolution of a self-impedance port after a power 2 Thermal Structure Functionstep input has been applied. It means that the temperature Thermal tructur e fronmust be measured where the heat is dissipated. This Thermal transients measured from real components inthermal transient can be measured from a real device or response to a unit power step can be represented as asimulated from a physical level model (finite elements, multiexponential transient.finite differences,...). The result is a curve which depicts n t

the heat flow map ofthe electronic component measured. T(t)=Z R1 K1-eTh +n(t) (1)The structure function is actually computed from the i=1 )

time-constant spectrum obtained from the step response where Ri are the amplitudes and x1 are the time-constants.through deconvolution techniques in frequency or time A edomain as Szekely [5] and Carmona [6] have respectively the emponents the sumteio nov iec an

shown. ~~~~~~~~~~~thereal component case the summation over i becomes anshown. integral because the time-constant spectrum is continuous.After discretizing the continuous time-spectrum we If we discretize the unknown continuous time-constant

obtain a Foster network (fig. 1) which is formed by the spectrum of a real component an equation like eq.1 can beseries connection of n-stages, where each stage is the obtained to represent the thermal step response of the realparallel connection of a RC pair. The amplitude of the component. Dividing eq. 1 by the power applied thediscretized time-constant spectrum becomes the R value, thermal impedance function, Z(t), is obtained. Theand the C value is determined through the known relation'x=RC, where x is the corresponding discretized time-

1]-4244-0276-X/06/$20. 00®)2006 IEEE -4-

7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006

Laplace transform of the thermal impedance function in precision format even the Cauer network resulting from athe Foster representation is: 20-stage network cannot be accurately computed. For this

n R reason, to compute Cauer networks with a few tens ofZF (S) Z (2) stages, longer mantissas are needed and special

i=1+ S -

i arithmetics which can deal with 50-100 significativewith lc=Ri Ci. Let us see which is the Laplace digits (depending on the number of stages in the Cauertransformation of the impedance function in the Cauer network) [4].type network: Once the Cauer network has been obtained, the

Zc (s) = + 1 structure function is a simple representation of the Cauer

s * C' + 1 network values which, we would like to remind that, haveRI'+ 1 (3) physical meaning. The structure function is the graphicalR1+ 1 representation of the cumulative capacitance versus the

s*C, + R' cumulative resistance of the Cauer network values. For2 *-- this reason some authors call it cumulative structure

expressed as a continued fraction expansion. function. It was first introduced by Protonotarios andSzekely's group has advocated that this discrete time- Wing [7] and it depicts the internal structure of a non-

constant spectrum may be obtained by a deconvolution uniform ID distributed RC line.procedure, followed by a discretization. However, it is An example of such a structure function is plotted inwell-known that deconvolution is an ill-posed problem, fig.2, where it is explicitly shown what is the significancebeing sensitive to noise and truncation. Moreover, most of the Cauer network RC values in the plot.deconvolution processes are not optimal in approximatingthe original measured signal. Alternatively, the so-calledExponential Series Method (ESM) may be used Thismethod fixes a time-constant distribution and fits theamplitudes through a least squares optimization process.The time-constant distribution used is typically a v C

logarithmic distribution with n time-constants for eachtime decade, so that the same weight is given to every /time decade. In a linear distribution case, too much R.importance would be given to the steady statetemperature. Once the amplitudes of eq. 1 have beenadjusted to reproduce the thermal transient we candirectly draw a Foster network (see section 1). Then, thenext step is to compute the so called Foster-Cauertransformation [4]. This iterative process is again an ill- ER (K/W)posed problem. The steps to extract the R' and C' values Figure 2: Structure function representation.from the R and C values start from converting eq.2 in arational expression expanding the summation: In a uniform slab the significance of the Cauer

z )an1I*Sn-I +a2.n-2 + +I*s5 +aOa0 network values can be directly related to material thermal'~F~ bn s' + bn s-l + +b, sl +bo properties (K, thermal conductivity, p, density, and C,Once we have two polynomials in s the extraction of specific heat capacity) and geometry parameters (length,

L, and cross sectional area, A) through the followingthe C'1 value iS made as follows: equtinsb ~~~~~~~~~~equations:

liY (s)_= s=s Cl_imYc(s) (5) LR= Ln-I icKA (7)

where Y(s) is the admittance function, the inverse of E = p *C * L * Athe impedance function Y(s)=1/Z(s). Then:the impedance funtion Y(s)= I/s)Then:If we consider the i-th value of 1R' represented as

C= (6) R'i, is the cumulative sum of the Cauer resistances fromn"-I

(6) the first stage to the i stage, and 7C'i is the cumulativesubstracting this capacitance from eq.4 the s' term will sum of the Cauer capacitances from the first stage to the i

vanish and the R'1 value will be determined as in eq.5. stage. If we consider a region where an homogenousThe first step, the expansion of eq.2 to obtain eq.4, is material is present and the cross-sectional area is constant

the more time consuming operation. This process must be a constant slope would appear in the structure function. Amade with many significative digits because the iterative change in the cross-sectional area would yield a change ofprocess of substracting the different eliminated elements slope in the plot. When two different materials arefrom eq.4 needs to be as exact as possible. If the Foster- connected the structure function will produce a change ofCauer transformation is computed with an IEEE double- slope as well. This are the basic properties of the structure

7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006

function, if the material properties of a device are known, surfaces except the bottom one, where Dirichlet boundaryit can predict the cross-sectional area, and if the cross- conditions will be applied, fixing the temperature.sectional area is constant it is possible to determine thenumber of material transitions and some relation between adiabatic surfacesthe different material properties.

The slope of the structure function can be calculatedas a function of the material properties and the devicegeometry as: L KC1

rt II2

R D. .A. (8)If this relation is different from one material to the fixed temperature

other, a change of slope will appear in the structurefunction. For this reason the differential structure function

Fiue4Unfnslbsdasreecem elowas defined [4], to simplify the slope change detection. It FingureL 4:d Uns-sciform lsabruedas reeec.mdlocontains the same information as the structure function lnt n rs-etoa raAbut it is presented in a different way.

,

In this representation, changes of slope are detected as Startion fote hea equatinranonsiderigae npeaks (fig.3) and they can mean a material change or a dcross-sectional area variation. l2T(x, t) AT(x, t)

where K is the thermal conductivity, p is the density,and C is the specific thermal heat capacitance. Applyingthe before mentioned boundary conditions and after somecalculations we end up with the following time-constantsand amplitudes [10]:

1nK n 2 2

8-L2TICAfln . /

where n tends to inf cnity, and A is the cross sectionalarea. This can be represented in time-constant spectrumplot as seen in fig.5.

Y-R (K/W)

Figure 3: Differential structure function representationwith the presence of peaks. 100

Once the properties of the structure function havebeen presented and the Foster-Cauer transformation hasbeen defined, let us face a real problem. Thermal transientmeasurements have noise (even if minimized by careful fmeasurements). Accurate determination of the time-constant spectrum is very noise sensitive, as well as theFoster-Cauer transformation. In consequence we expectthat a precise determination of differential structurefunction will require extremely low signal-to-noise ratios 1o-4in the thermal impedance function. The results of this 10-4 10-3 (-2)study will be shown in section 4. °(sFigure 5: Uniform slab time-constant spectrum.3. Reference Model

Let us define our reference model which will be used As we know from section 2, the resulting structurefor testing purposes. In order to calculate analytically the function for this uniform slab must be a constant valuetime-constant spectrum a simple geometry will be equal to 1 for K=p=C=A=1. If we compute the theoreticalconsidered [8-10]. The simplest case is an uniform slab of differential structure function truncating the infinite seriesfinite length, L, and constant cross-sectional area, A, see of time-constants at 10, 30 and 50 terms respectively, thefig.4. The boundary conditions applied are adiabatic to all results shown in fig.6 are obtained. As it can be seen the

7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006

more terms we consider the more similar is the time-constant distribution is show in fig.9. And thedifferential structure function to a constant value of 1. resulting differential structure function is plotted in fig. 10.

The Foster-Cauer transform has been made with aprecision of 100 decimal digits, through a variable

,------- - - - - - ------ 1 precision arithmetic implemented in Matlab..

0.8 -simulatedfiltered

_ 0.6 00.8

0.4

10 terms 2 0.6

---30 terms0.2

-50 terms E0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Rth [K/W]

Figure 6: Differential structure functions for the 0.2 -

uniform slab with 10, 30 and 50 terms.

4. Sensitivity Analysis 0 - -3 - l 0*r 1 1 * 10~~~~~i-5 10-4 10-3 io-2 i-10-1 10°l1In order to study the influence of thermal transient time (s)

noise in the computation of the structure function, we Figure 7: Simulated thermal transient with an additivehave used the reference model presented in the previous gaussian noise of variance G2=l0-5 K2 and thesection to test the results of our computations, because we gaussiandnoiserof vrane o 1tainnd hknow the time-constant spectrum of the uniform slab, andwe know its theoretical structure function. Our io-5methodology will be creating thermal transients withdifferent levels of noise, and setting an initial logarithmic 106distribution of time-constants ranging the whole timevector during simulation, and adjusting through a least-

7

squares algorithm the amplitudes of these time-constants 10

to fit the different noisy thermal transients. Once these .\amplitudes have been adjusted we can apply the Foster- 0L 10Cauer transform, and plot the corresponding differential . -structure functions comparing them with the theoretical 10-differential structure function shown in fig.6.

An important step which must be considered is that 10

real measurements are usually made with a constantsampling time. It means that most of the measurements 10o11 4 0correspond to the steady state measurement. For this 10-5 10-4 10-3 tio1-() - 10° 101reason a logarithmic time-scale transformation is Figure 8: Noise power after filtering thermal transientpreferred, reducing simultaneously the noise level.. The of fig.7 to a logarithmic time-scale with 10 points peroptimal procedure to carry-out this transformation has decade.been studied by the authors [11].A thermal transient polluted with noise is plotted in The 10 different sets of adjusted amplitudes plotted in

fig.7 corresponding to the uniform slab transient retaining fig.9, show the effect of the exponentials non-the first 50 terms in the time-constant spectrum plus an orthogonality, specially seeing the wider spreading ofadditive gaussian noise of variance a =l0 K, sampled at amplitudes for small time-constant values. When a hightS=10-5s. The filtered thermal transient with a logaritmic density of time-constants are present, only one of thetime-scale (10 points per decade) is plotted as well. amplitudes has a value higher than the theoretical which

The resulting noise power after filtering is plotted in produces lower values in the neighbouring time-constants.fig.8, where we can see the dramatic noise reduction for Moreover, another effect must be taken into account inlarge values of time, where many measurements are this plot, the higher the theoretical amplitude the betteravailable in the linear time-scale. the adjusted value. This happens because the sensitivity of

As a first case, we will apply the least squares this amplitude to the output is maximum. When smalleroptimization process to a time-constant distribution values are being adjusted their influence on the output iscorresponding to the theoretical one truncated to 20 terms not much strong and so their value is not enough correctlyand with an additive noise with G2=10-5 K2. The resulting fitted. On the other hand amplitudes corresponding to

-4-7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006

small time-constants have a wider spreading around the exponent of the amplitudes are the adjusted parameters.theoretical value. The resulting differential structure This helps the algorithm to determine low amplitudefunctions shown in fig. 10 are not satisfactory from a values.quantitative point of view because relative errors around The amplitude distribution for 10 different adjusted100% are present in some parts of the differential amplitudes for G2=1 0-6 K2 case is plotted in fig.11. As westructure function, showing a high sensitivity to the can see, big amplitudes which correspond to large timethermal transient noise. values are the best fitted ones. This is the expected result,

100 since this part of the transient features the best signal-to-noise ratio. The resulting differential structure functionsare plotted in fig.10.

102 1

~~~~~~~~~~~~ ~~~~~~~~~~~102

102~~~~~~~~~~~~~~ 10

1 o~ . 0 t7-1 100 o1

K.12) obtained fixing the time-constants to the theoretical 1 10 - 102 1-1 100 101values. T(S)

10 -Figure 11: Adjusted time-constant spectrums of 109 different filtered transients with additive gaussian noise

(&2=106 K2) compared to time-constant spectrum of the8 - uniform slab (black circles).

7 ~~~~~~~~~~~~~~~~~~~~~10

4~~~~~~~~~~~~~~~~~~~~~~~~~

1 7

4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C-)~~~~~~~~~~~~~~~~~~~~~~~~~~~o

1 0-5~~~~~~~~~~~~~~

°0 01 02 03 04 0. 06 07 08 0. 1 2-R(W)

Figure 10: Differential structure function obtained (10 '6cases with G2=10-5 K2) after the amplitude optimization 00, 0.1 0.2 0.3 0.4 0.5 0., . . 0.9 1and fixing the time-constants to the theoretical values. 10 (W)

Let us now consider a case where we distribute thetime-constants in a logarithmic scale with 20 points per 8decade from t-10-5s to t-lOs resulting in 120 time- 7_constants. Each time-constant has an amplitude which e6must be adjusted to fit each of the filtered thermal X tfranslens.

Different levels of noise have been considered, which oare 10-6, 4106, 10-, 410-5 and i0-4 K2. For each noiselevel 10 random noisy transients have been generated and 2 Mfiltered to obtain the logarithmnic timec-scale transient. For 1Nr .*-_^N--^f-s@

time-constants have been adjusted to fit this transient. In 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1fact, to improve the least squares fitting process, the ER' (K/W)

7th.-.o:nhm,enaaMlpssmaoaEpiniMrEcocaMrSt

7th7t ofo hra,Mcaia n utpysc iuainadEprmnsi ir-letoisadMcoSses uoiE20

10_ In fact, we have found that this incorrect behaviour is9- due to errors in the time-constant spectrum, in particular8_ in the position of the time-constants. When these incorrect7 time-constants are present their amplitude is adjusted to a

low value, and it is the cause for the strong distortions in< 6- _the computed differential structure function.

a,13: s- This behaviour will appear whenever these small4-4 values are present, and as we cannot know in advance

which is the theoretical time-constant distribution from a3-A

given thermal transient, the differential structure function2

^01will always present this incorrect behaviour. In any case,

i __rand even if the divergences have been eliminated the0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [thermal strucrure function remains quite noisy. This is

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1YR' (KIW) important to notice, since in the first case we are10 assuming that we know exactly the time constant position91 1 t§gg and the only introduced error comes from the limited8 precision in the estimation of the amplitudes.7 || Conclusions6-6 The computlation of the thermal structure function

from thermal impedance transients is a complex andt- g,1ls IIDi sensitive signal processing operation. Severall ill-posed

steps appear like the estimation of the time-constant3011001|l h 11§Z spectrum and the Foster-Cauer transformation. We have2 lt2\^/\A -\ 1^ zU 11 found that very limited noise power in the thermal

transient leads to high noise levels in the thermal structurefunction. On the other hand, we have proved that the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 usual divergence of the differential thermal structure10 YR (KIW) function for high cumulative thermal resistances is not

related to the ambient influence, but is a purelymathematical artifact due to the limited precision in the

8 determination of the time-constant spectrum. Finally, it is7 also important to remark that the Foster-Cauer

transformation has to be performed with extendedarithmetics to avoid numerical problems for networks

_ 50|11|1 -lh IU g with a moderate number of stages.4-

Acknowledgments3|||011 ;1 Ath 11 I This work has been partially funded by the Spanish

2 I A Ministry of Education and Science, under project1 ~~~~~~~~~~~~Tec2004-07853-c02-0 1

o~~~~~~~~~~~~References0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1YR' (KIW)

Figure 12: Differential structure functions . .corresponding to (top to down) 21-6 410-6, 1-5 1 1. De Wolf, I., Rasras, M., "From photon emssonand 10 K and the theoretical result (black circles). microscopy to Raman spectroscopy: Failure analysis

in microelectronics," Eur. Phys. J. Appl. Phys., Vol.27 (2004), pp. 59-65.

It is clear the influence of the noise power in the 2. De0Wol, I. " F r l M o2. De Wolf, I., "New Failure Analysis Methods indifferential structure functions evaluated. The results Microelectronics," Proc 4th EuroSIME Conf, Aix-en-obtained are quite noisy and deviations exceeding 100o Provence, France, May. 2003, pp. 367-373.the theoretical value are present, specially in the initial 3. Microelectronics Failure Analysis, Desk Referenceand final part.

An important characteristic is the behaviour for large Fifth Edition, (ASM Interational, 2004).I VIT- 1-- --, , I 4. Sze'kely, V., Van Bien, T., "Fine Structure of heatvalues of ER', which always corresponds to huge values 'zkey V.,Va' in . Fn tutr fha

of C'/'.Thi ha bee obere by ote auhr an*i flow path in semiconductor devices: a measurementha bee atrbue to th inlec of th ambien and identification method," Solid State Electonics,surrounding the component. In this case, this explanation Vo.3,N.9(8),p.16-6.can be completely excluded since the model does notconsider the ambient.

-6-7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006

5. Szekely, V., "Indentification of RC Networks byDeconvolution: Chances and Limits," IEEE Trans.Circ. Syst., Vol. 45, No. 3 (1998), pp. 244-258.

6. Carmona, M., Marco, S., Palacin, J., and Samitier, J.,"A time-domain method for the analysis of thethermal impedance response preserving theconvolution form," IEEE Transactions onComponents Packaging and ManufacturingTechnology Part A, Vol. 22 (1999), pp. 23 8-244.

7. Protonotarios, E.N., Wing, O., "Theory of nonuniformRC lines", IEEE Trans. Circ. Theory, Vol. 14, No. 1(1967), pp. 2-12.

8. Szekely, V., "On the representation of Infinite-LengthDistributed RC One-Ports," IEEE Trans. Circ. Syst.,Vol. 38, No. 7 (1991), pp. 711-719.

9. Codecasa, L., "Canonical Forms of One-Port PassiveDistributed Thermal Networks," IEEE Trans. Comp.Pack. Tech., Vol. 28, No. 1 (2005), pp. 5-13.

10. Ghausi, M.S., Kelly, J.J., Introduction to DistributedParameter Networks, Holt, Rinehart and Wilson (NewYork, 1968).

11. Palacin, J., Marco, S., Samitier, J., "SuboptimalFiltering and Nonlinear Time Scale Transformationfor the Analysis of Multiexponencial Decays", IEEETrans. Instrum. Meas., No. 50 (2001), pp. 135-140.

7th. Int. Conf: on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006