Microelectronics Reliability 48 (2008) 1837–1846

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Microelectronics Reliability

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Finite element model verification for packaged printed circuit boardby experimental modal analysis

Ying-Chih Lee a,b, Bor-Tsuen Wang c, Yi-Shao Lai a, Chang-Lin Yeh d, Rong-Sheng Chen b,*

a Central Labs, Advanced Semiconductor Engineering, Inc., Kaohsiung 811, Taiwanb Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwanc Department of Mechanical Engineering, National Pingtung University of Science and Technology, Pingtung 912, Taiwand Shanghai Engineering Center, Advanced Semiconductor Engineering, Inc., Shanghai 201203, China

a r t i c l e i n f o

Article history:Received 28 March 2008Received in revised form 22 July 2008Available online 12 September 2008

0026-2714/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.microrel.2008.07.068

* Corresponding author. Tel.: +886 6 2757575x633E-mail address: rschen@mail.ncku.edu.tw (R.-S. Ch

a b s t r a c t

In this work, the experimental modal analysis (EMA) was performed to establish an equivalent finite ele-ment (FE) model for a standard Joint Electron Device Engineering Council (JEDEC) drop test printed circuitboard (PCB) mounted with packages in a full array. Material properties of the equivalent FE model of thepackaged PCB were calibrated through an optimization process with respect to natural frequencies basedon EMA results obtained with a free boundary condition. The model was then applied to determinescrewing tightness of the packaged PCB corresponding to a fixed boundary condition with the four cor-ners of the PCB constrained, as defined by JEDEC for a board-level drop test. Modal damping ratios of thepackaged PCB were also provided.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The integrity of solder joints in electronic packages under med-ium to harsh dynamic loading environments has become a criticalissue with the prevalence of portable electronic devices as well asthe introduction of stiff and brittle Pb-free solder alloys [1,2]. Sev-eral accelerated board-level reliability testing standards aiming atevaluating solder joint reliability that corresponds to these dy-namic loading environments, such as cyclic bend [3,4], drop impact[5,6], or vibration [7–12], have been proposed by the Joint ElectronDevice Engineering Council (JEDEC) [13–16] and have been fol-lowed by the industry in qualifying the products.

For the design purpose, the finite element analysis (FEA) has longbeen proven to be conducive in selecting proper structural configu-rations and materials for electronic packages without the need ofcostly and time consuming experiments. However, the accuracy ofnumerical solutions depends greatly on the feasibility of modelingthat includes proper settings of geometry, boundary and loadingconditions, and material properties. For FEA of a board-level test,modeling of the printed circuit board (PCB) is generally consideredas the source that brings the most uncertainty to the numericalsolutions. Overall mechanical properties of the PCB can vary accord-ing to different numbers of metal layers, different layouts of circuits,and different polymeric and composite reinforcing materials used inthe fabrication. Tiny circuits and their complex layouts inside thePCB also limit the possibility of comprehensive modeling of the PCB.

ll rights reserved.

28; fax: +886 6 2766549.en).

We note, however, if the concern is only with overall mechani-cal properties of the PCB or the board- or system-level test vehiclethat contains PCB and packages mounted onto it, an experimentalmodal analysis (EMA) [17] can practically determine modal param-eters, including natural frequencies, mode shapes, and damping ra-tios, of the specific PCB or board- or system-level test vehiclewithout the need of prerequisite information of material propertiesand layouts of its individual constituent components [18–20]. Anequivalent finite element (FE) model calibrated by modal parame-ters obtained from EMA can thus be used with more flexibility forthe design purpose [5,21].

In the present work, we followed EMA to characterize modalparameters of a standard 132 � 77 � 1 mm JEDEC drop test board[13,15] mounted with 13 � 13 mm packages in a 3 � 5 full array,as shown in Fig. 1. The packaged PCB arranged with free and fixedboundary conditions are shown in Fig. 2. As depicted by the proce-dure in Fig. 3, material properties of the equivalent FE model of thepackaged PCB were calibrated through an optimization processbased on EMA results obtained with a free boundary condition.The model was then applied to determine screwing tightness ofthe packaged PCB corresponding to a fixed boundary conditionwith the four corners of the PCB constrained, as defined by JEDECfor a board-level drop test [13,15].

2. EMA and FEA

Experimental setups for EMA on the packaged PCB with free andfixed boundary conditions have been shown in Fig. 2. To avoidinterference, hammering was performed on the reversed side of

Fig. 1. Schematic of packaged PCB (not to the scale).

Fig. 3. Procedure for equivalent FE model establishment.

Fig. 4. Grid of measurement points (Accelerometer locations: N for free; j forfixed).

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the packaged surface, Fig. 4, in which the locations where theaccelerometer was affixed are labeled. For the free boundary con-dition, we measured hammering force and acceleration responsescorresponding to hammering locations in a 17 (vertical) � 29 (hor-izontal) array on the surface, shown by the black dots in Fig. 4. Forthe fixed boundary condition, hammering force and accelerationresponses corresponding to 88 hammering locations along the A1through A4 lines shown in Fig. 4 were measured. The frequency re-sponse functions (FRFs) between measured hammering force andacceleration responses were then curve-fitted to extract experi-mental modal parameters of the packaged PCB.

Fig. 5 shows full three-dimensional FE models built for thepackaged PCB with free and fixed boundary conditions. The modelfor the packaged PCB contained 12,335 linear hexahedral solid ele-ments along with 24 mass elements to take into account the accel-erometer mass. For the fixed boundary condition, 192 additionallinear spring-damper elements were employed to account for thetightness of screws at the four corners; the torque applied to tight-en the screws was 8 kgf-m. The damping effect of the screws wasneglected in this study while only the spring effect was considered.The spring constant was assumed to be constant and was to bedetermined in the subsequent optimization procedure after thematerial properties of the packaged PCB were determined fromthe optimization based on EMA and FEA results for the free bound-ary condition.

Initial material properties extracted from Yeh and Lai [22],listed in Table 1, were specified to PCB and packages. We assumethat the packages are isotropic while the PCB is cubic with three

Fig. 2. Packaged PCB with free and fixed boundary conditions (packages on reversed side).

Fig. 5. FE models of packaged PCB.

Table 1Initial material properties of packaged PCB [22]

PCB Package

Young’s modulus (GPa) 15.50 28.00Poisson’s ratio 0.20 0.35Mass density (g/cm3) 1.91 1.81

Table 2Convergence analysis with respect to natural frequencies (Unit: Hz)

Cell A B C D B vs. C

Number of elements

3620 7245 12,335 17,415 Relative error (%)

Mode1 178.97 148.91 147.34 145.48 1.0542 220.50 191.13 189.46 188.12 0.8753 465.73 393.58 386.87 385.85 1.7054 491.95 394.27 391.00 388.19 0.8295 495.89 401.51 395.20 388.39 1.5726 772.80 562.22 551.21 549.91 1.958

Table 3Comparison between initial and optimized material properties of packaged PCB

Initial Optimized Discrepancy (%)

Eb (GPa) 15.50 9.42 39.2Gb (GPa) 6.46 3.12 51.6mb 0.20 0.25 25.0qb (g/cm3) 1.91 2.05 6.8Ep (GPa) 28.00 20.00 28.6mp 0.35 0.4 14.3qp (g/cm3) 1.81 1.84 1.6

Fig. 6. Representative FRFs and corresponding coherence fun

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independent elastic constants, albeit isotropy was given to the PCBas an initial setting. The accelerometer mass is 1.5 g. A numericalconvergence analysis based on the initial material properties wascarried out with respect to natural frequencies of the first fivemodes of the packaged PCB, as shown in Table 2. Clearly fromthe results, the FE model with 12,335 elements has achieved suffi-cient accuracy with moderate computational efficiency.

An optimization process based on the coincidence of numericaland experimental natural frequencies was followed to iterate

ctions for packaged PCB with free boundary condition.

Table 4Natural frequencies and corresponding mode shapes for packaged PCB with free boundary condition

Mode Frequency (Hz) EMA FEA MAC(m,n)Discrepancy (%)

E1 vs. F1 161 vs. 154 0.914.3 (3,1)

E2 vs. F2 187 vs. 200 0.837.0 (3,1)

E3 vs. F3 356 vs. 357 0.910.3 (3,2)

E4 vs. F4 461 vs. 457 0.680.9 (4,1)

E5 vs. F5 492 vs. 467 0.705.1 (1,3)

E6 vs. F6 582 vs. 574 0.971.4 (2,3)

E7 vs. F7 660 vs. 684 0.973.6 (4,2)

E8 vs. F8 858 vs. 857 0.971.2 (5,1)

E9 vs. F9 913 vs. 921 0.940.9 (3,3)

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Table 4 (continued)

Mode Frequency (Hz) EMA FEA MAC(m,n)Discrepancy (%)

E10 vs. F10 1050 vs. 1075 0.922.4 (5,2)

E11 vs. F11 1220 vs. 1192 0.722.3 (4,3)

E12 vs. F12 1300 vs. 1278 0.751.7 (1,4)

E13 vs. F13 1400 vs. 1349 0.683.6 (2,4)

F14 1429 N/A N/AN/A (4,4)

E15 vs. F15 1590 vs. 1596 0.930.4 (6,1)

E16 vs. F16 1710 vs. 1739 0.781.7 (5,3)

E17 vs. F17 1740 vs. 1759 0.841.1 (3,4)

E18 vs. F18 1980 vs. 1974 0.910.3 (6,2)

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Fig. 7. Representative FRFs and corresponding coherence functions for packaged PCB with fixed boundary condition.

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material properties as well as the spring constant of the packagedPCB. For the free boundary condition, design variables includeYoung’s modulus (Eb), shear modulus (Gb), Poisson’s ratio (mb),and mass density (qb) of the PCB as well as Young’s modulus(Ep), Poisson’s ratio (mp), and mass density (qp) of the packages.The design variables were than separately optimized in the orderof Young’s modulus, shear modulus, Poisson’s ratio, and mass den-sity with respect to the minimal sum of square errors betweennumerical and experimental natural frequencies and the criterionwas set at a relative error within 10%. The optimization followedthe sub-problem approximation method provided by the ANSYSsoftware package (ANSYS, Inc., Canonsburg, PA, USA) with randomdesign generation and sweep generation. After the optimal mate-rial properties were obtained for the packaged PCB using resultscorresponding to the free boundary condition, the spring constantwas then solely determined following the same optimization pro-cess based on numerical and experimental natural frequencies cor-responding to the fixed boundary condition.

Material properties of individual components can certainly bemeasured independently. However, since in this work the compo-nents were modeled in a compact manner that neglected all thesmall constituents in order to reduce the problem size, the opti-mized overall material properties of individual components shouldmore or less involve structural interactions between the compo-nents. As a result, the optimized material properties of individualcomponents should be regarded as parameters corresponding tothe specific modeling in this work, and may not be directly compa-rable with those from independent measurements.

3. Results and discussion

3.1. Free boundary condition

For the free boundary condition, we measured hammering forceand acceleration responses corresponding to 493 hammering loca-

tions on the surface of the packaged PCB. Hammering was per-formed three times at each of the 493 locations. ExperimentalFRFs corresponding to individual hammering locations werecurve-fitted using ME’scope VES v. 3.0 (Vibrant Technology, Inc.,Scotts Valley, CA, USA) to obtain experimental modal parametersof the packaged PCB, whose material properties were then opti-mized with respect to natural frequencies and listed in Table 3.The measure of anisotropy of the PCB is 2Gb(1 + mb)/ Eb = 82.8% withthe optimized material properties. Our study also showed that noeligible correlation could be obtained in the present case whenthe PCB was assumed to be isotropic.

We denote a pair (i, j) in which i stands for the output signallocation, i.e., the accelerometer location and j stands for the inputsignal location, i.e., the hammering location. In the present case,the accelerometer located close to the 465th hammering point.Fig. 6 shows two examples of FRFs up to 2 kHz and correspondingcoherence functions for (465,180) and (465,465), for which the lat-ter indicates that the hammering location is right beside the accel-erometer. In the figure, the synthesized curve refers to the FRF aftercurve-fitting based on the multiple-mode method to compensatelimited signal resolution so that modal parameters can be properlyestimated; the theoretical curve represents the FE solution basedon the optimized material properties and measured damping ofthe packaged PCB, which is summarized in Section 3.3. From thefigure we note that measured and numerical FRFs are in reasonablygood agreement. Moreover, the coherence functions approachunity up to 2 kHz except at the frequencies where anti-resonance[17] occurs, indicating that at each hammering location, the threehammerings generated consistent results and hence the experi-ments were reliable.

To further verify the equivalent FE model with the optimizedmaterial properties, we compare its natural frequencies and corre-sponding mode shapes up to 2 kHz with EMA results using themode assurance criterion (MAC) [17], as shown in Table 4. In thetable, En and Fn refer to EMA and FEA results, respectively, for the

Table 5Natural frequencies and corresponding mode shapes for packaged PCB with fixed boundary condition

Mode Frequency (Hz) EMA FEA MAC(m,n)Discrepancy (%)

E1 vs. F1 194 vs. 192 0.991.3 (1,1)

E2 vs. F2 364 vs. 380 0.994.3 (1,2)

E3 vs. F3 487 vs. 491 0.960.9 (2,1)

E4 vs. F4 549 vs. 553 0.950.7 (3,3)

E5 vs. F5 695 vs. 696 0.940.1 (4,1)

E6 vs. F6 833 vs. 850 0.662.0 (4,2)

E7 vs. F7 845 vs. 860 0.841.7 (3,3)

E8 vs. F8 1100 vs. 1100 0.930.0 (5,1)

(continued on next page)

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Table 5 (continued)

Mode Frequency (Hz) EMA FEA MAC(m,n)Discrepancy (%)

E9 vs. F9 1290 vs. 1290 0.800.0 (4,3)

E10 vs. F10 1320 vs. 1311 0.890.7 (5,4)

E11 vs. F11 1500 vs. 1448 0.453.5 (5,4)

F12 1517 N/A N/AN/A (3,2)

E13 vs. F13 1670 vs. 1699 0.401.7 (5,4)

E14 vs. F14 1750 vs. 1757 0.580.4 (4,4)

E15 vs. F15 1880 vs. 1874 0.460.3 (5,2)

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n-th mode. Modal characteristics along longitudinal and lateraldirections are shown in parenthesis. The MAC, defined by

MACðf/xg; f/pgÞ ¼jf/xg

Tf/�pgj2

f/xgTf/�xgf/pg

Tf/�pg; ð1Þ

is frequently used to quantitatively evaluate the discrepancy be-tween measured mode shape, {/x}, and predicted mode shape,

{/p}. The asterisk in Eq. (1) denotes the conjugate. For two coinci-dent mode shapes, MAC = 1.

It is clear from Table 4 that there are 18 modes up to 2 kHz forthe packaged PCB with the free boundary condition. The EMA re-sult for the 14th mode could not be obtained because the acceler-ometer may locate at a nodal point of this particular mode. Forremaining modes, natural frequencies obtained by EMA and FEA

Table 6Modal damping ratios of packaged PCB

Mode Free boundary Fixed boundary

Experimental (%) Accumulated average (%) Experimental (%) Accumulated average (%)

E1 2.139 2.139 1.432 1.432E2 0.577 0.939 0.698 1.065E3 1.300 1.339 1.693 1.274E4 0.541 1.139 1.658 1.370E5 0.475 1.006 0.777 1.252E6 0.675 0.951 0.772 1.172E7 0.620 0.904 3.153 1.455E8 0.840 0.896 0.574 1.345E9 0.549 0.857 0.707 1.274E10 0.722 0.844 0.888 1.235E11 0.675 0.828 0.865 1.202E12 0.518 0.803 0.668 1.157E13 0.563 0.784 0.759 1.126E14 0.760 0.782 0.928 1.112E15 0.577 0.769E16 0.601 0.758E17 1.251 0.787

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are in good agreement with discrepancies within 7%. Mode shapesfrom EMA and FEA also agree very well with each other with MACsgreater than 0.9 except for only a few modes whose MACs aresmall. The MACs are particularly small for the 4th and 5th modes,whose natural frequencies are nearly indistinguishable at around620 Hz for the bare packaged PCB with no accelerometer attached.Clearly, for these two close modes, the simplification of the accel-erometer by mass elements was only able to capture their modeshapes qualitatively.

3.2. Fixed boundary condition

Following the same grid numbering as in the case of the freeboundary condition, the accelerometer in the case of the fixedboundary condition located close to the 468th point. Representa-tive FRFs for (468,74) and (468,358) up to 2 kHz are shown inFig. 7. Clearly, measured and numerical FRFs are in reasonablygood agreement while the coherence functions approach the unityexcept at the frequencies where anti-resonance occurs. With theoptimized material properties, the spring constant for the pack-aged PCB with the fixed boundary condition was determined as2.78 MN/m through the optimization process.

Table 5 summarizes and compares natural frequencies and cor-responding mode shapes up to 2 kHz obtained by EMA and FEA.The comparison of mode shapes is shown along lines A1 throughA4 because hammering was performed along these lines only tospeed up the experiment. It is clear there are 15 modes up to2 kHz for the packaged PCB with the fixed boundary condition.The EMA result for the 12th mode could not be obtained becausethe accelerometer may locate at a nodal point of this particularmode. For remaining modes, natural frequencies obtained byEMA and FEA are in very good agreement with discrepancies lessthan 2%. Mode shapes from EMA and FEA also agree very well witheach other with MACs greater than 0.9 except for only a few modeswhose MACs are small.

3.3. Modal damping ratios

Modal damping ratios can only be obtained from experimentalmeasurements. In Table 6 we show modal and accumulated aver-age damping ratios for the packaged PCB with free and fixedboundary conditions. The accumulated average damping ratio isdefined by

�n ¼Pn

i¼1ni

n; ð2Þ

where ni represents the damping ratio of the i-th mode. Since mostof the software packages do not allow the input of modal dampingratios, the accumulated average damping ratio stands for an essen-tial structural system characteristic in practice. From the table wenote that in general a higher mode has a smaller damping ratio.These damping ratios are beneficial to the transient analysis ofthe packaged PCB that follows the concept of modal superposition[23,24], although this linear superposition concept can only be re-garded as a qualitative estimation under severe dynamic loadingconditions that bring in significant structural nonlinearity in the re-sponses of the packaged PCB.

4. Conclusions

An equivalent FE model for a standard 132 � 77 � 1 mm JEDECdrop test PCB mounted with 13 � 13 mm packages in a 3 � 5 fullarray has been established based on EMA results correspondingto free and fixed boundary conditions. Natural frequencies andmode shapes obtained by EMA and FEA are generally in very goodagreement. We also provide modal damping ratios obtained exper-imentally, which would benefit the transient analysis of the pack-aged PCB that follows the concept of modal superposition. As aconcluding remark, we have demonstrated in this work that overallmaterial properties and screwing tightness of a packaged PCB canbe determined by incorporating EMA and FEA along with an opti-mization process. This technique is conducive to the FE modelingof complex PCB structures for the design for mechanical reliability.

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