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Finite element model verification for packaged printed circuit board by experimental modal analysis
Microelectronics Reliability 48 (2008) 1837–1846
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Finite element model verification for packaged printed circuit board by 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, Taiwan Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan c Department of Mechanical Engineering, National Pingtung University of Science and Technology, Pingtung 912, Taiwan d Shanghai Engineering Center, Advanced Semiconductor Engineering, Inc., Shanghai 201203, China b
a r t i c l e
i n f o
Article history: Received 28 March 2008 Received in revised form 22 July 2008 Available online 12 September 2008
a b s t r a c t In this work, the experimental modal analysis (EMA) was performed to establish an equivalent finite element (FE) model for a standard Joint Electron Device Engineering Council (JEDEC) drop test printed circuit board (PCB) mounted with packages in a full array. Material properties of the equivalent FE model of the packaged PCB were calibrated through an optimization process with respect to natural frequencies based on EMA results obtained with a free boundary condition. The model was then applied to determine screwing tightness of the packaged PCB corresponding to a fixed boundary condition with the four corners of the PCB constrained, as defined by JEDEC for a board-level drop test. Modal damping ratios of the packaged PCB were also provided. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The integrity of solder joints in electronic packages under medium to harsh dynamic loading environments has become a critical issue with the prevalence of portable electronic devices as well as the introduction of stiff and brittle Pb-free solder alloys [1,2]. Several accelerated board-level reliability testing standards aiming at evaluating solder joint reliability that corresponds to these dynamic loading environments, such as cyclic bend [3,4], drop impact [5,6], or vibration [7–12], have been proposed by the Joint Electron Device Engineering Council (JEDEC) [13–16] and have been followed by the industry in qualifying the products. For the design purpose, the finite element analysis (FEA) has long been proven to be conducive in selecting proper structural configurations and materials for electronic packages without the need of costly and time consuming experiments. However, the accuracy of numerical solutions depends greatly on the feasibility of modeling that includes proper settings of geometry, boundary and loading conditions, and material properties. For FEA of a board-level test, modeling of the printed circuit board (PCB) is generally considered as the source that brings the most uncertainty to the numerical solutions. Overall mechanical properties of the PCB can vary according to different numbers of metal layers, different layouts of circuits, and different polymeric and composite reinforcing materials used in the fabrication. Tiny circuits and their complex layouts inside the PCB also limit the possibility of comprehensive modeling of the PCB.
* Corresponding author. Tel.: +886 6 2757575x63328; fax: +886 6 2766549. E-mail address: [email protected] (R.-S. Chen). 0026-2714/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2008.07.068
We note, however, if the concern is only with overall mechanical properties of the PCB or the board- or system-level test vehicle that contains PCB and packages mounted onto it, an experimental modal analysis (EMA) [17] can practically determine modal parameters, including natural frequencies, mode shapes, and damping ratios, of the specific PCB or board- or system-level test vehicle without the need of prerequisite information of material properties and layouts of its individual constituent components [18–20]. An equivalent finite element (FE) model calibrated by modal parameters obtained from EMA can thus be used with more flexibility for the design purpose [5,21]. In the present work, we followed EMA to characterize modal parameters 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 fixed boundary conditions are shown in Fig. 2. As depicted by the procedure in Fig. 3, material properties of the equivalent FE model of the packaged PCB were calibrated through an optimization process based on EMA results obtained with a free boundary condition. The model was then applied to determine screwing tightness of the packaged PCB corresponding to a fixed boundary condition with the four corners of the PCB constrained, as defined by JEDEC for a board-level drop test [13,15].
2. EMA and FEA Experimental setups for EMA on the packaged PCB with free and fixed boundary conditions have been shown in Fig. 2. To avoid interference, hammering was performed on the reversed side of
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Fig. 1. Schematic of packaged PCB (not to the scale).
the packaged surface, Fig. 4, in which the locations where the accelerometer was affixed are labeled. For the free boundary condition, we measured hammering force and acceleration responses corresponding to hammering locations in a 17 (vertical) 29 (horizontal) array on the surface, shown by the black dots in Fig. 4. For the fixed boundary condition, hammering force and acceleration responses corresponding to 88 hammering locations along the A1 through A4 lines shown in Fig. 4 were measured. The frequency response functions (FRFs) between measured hammering force and acceleration responses were then curve-fitted to extract experimental modal parameters of the packaged PCB. Fig. 5 shows full three-dimensional FE models built for the packaged PCB with free and fixed boundary conditions. The model for the packaged PCB contained 12,335 linear hexahedral solid elements along with 24 mass elements to take into account the accelerometer mass. For the fixed boundary condition, 192 additional linear spring-damper elements were employed to account for the tightness of screws at the four corners; the torque applied to tighten the screws was 8 kgf-m. The damping effect of the screws was neglected in this study while only the spring effect was considered. The spring constant was assumed to be constant and was to be determined in the subsequent optimization procedure after the material properties of the packaged PCB were determined from the optimization based on EMA and FEA results for the free boundary condition. Initial material properties extracted from Yeh and Lai [22], listed in Table 1, were specified to PCB and packages. We assume that the packages are isotropic while the PCB is cubic with three
Fig. 3. Procedure for equivalent FE model establishment.
Fig. 4. Grid of measurement points (Accelerometer locations: N for free; j for fixed).
Fig. 2. Packaged PCB with free and fixed boundary conditions (packages on reversed side).
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846
Fig. 5. FE models of packaged PCB.
Table 3 Comparison between initial and optimized material properties of packaged PCB
Table 1 Initial material properties of packaged PCB [22]
Young’s modulus (GPa) Poisson’s ratio Mass density (g/cm3)
PCB
Package
15.50 0.20 1.91
28.00 0.35 1.81
Eb (GPa) Gb (GPa)
mb qb (g/cm3) Ep (GPa)
mp qp (g/cm3)
Table 2 Convergence analysis with respect to natural frequencies (Unit: Hz) Cell
A
B
C
D
B vs. C
Relative error (%)
Number of elements
Mode 1 2 3 4 5 6
3620
7245
12,335
17,415
178.97 220.50 465.73 491.95 495.89 772.80
148.91 191.13 393.58 394.27 401.51 562.22
147.34 189.46 386.87 391.00 395.20 551.21
145.48 188.12 385.85 388.19 388.39 549.91
1.054 0.875 1.705 0.829 1.572 1.958
Initial
Optimized
Discrepancy (%)
15.50 6.46 0.20 1.91 28.00 0.35 1.81
9.42 3.12 0.25 2.05 20.00 0.4 1.84
39.2 51.6 25.0 6.8 28.6 14.3 1.6
independent elastic constants, albeit isotropy was given to the PCB as an initial setting. The accelerometer mass is 1.5 g. A numerical convergence analysis based on the initial material properties was carried out with respect to natural frequencies of the first five modes of the packaged PCB, as shown in Table 2. Clearly from the results, the FE model with 12,335 elements has achieved sufficient accuracy with moderate computational efficiency. An optimization process based on the coincidence of numerical and experimental natural frequencies was followed to iterate
Fig. 6. Representative FRFs and corresponding coherence functions for packaged PCB with free boundary condition.
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Table 4 Natural frequencies and corresponding mode shapes for packaged PCB with free boundary condition Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
MAC (m,n)
E1 vs. F1
161 vs. 154 4.3
0.91 (3,1)
E2 vs. F2
187 vs. 200 7.0
0.83 (3,1)
E3 vs. F3
356 vs. 357 0.3
0.91 (3,2)
E4 vs. F4
461 vs. 457 0.9
0.68 (4,1)
E5 vs. F5
492 vs. 467 5.1
0.70 (1,3)
E6 vs. F6
582 vs. 574 1.4
0.97 (2,3)
E7 vs. F7
660 vs. 684 3.6
0.97 (4,2)
E8 vs. F8
858 vs. 857 1.2
0.97 (5,1)
E9 vs. F9
913 vs. 921 0.9
0.94 (3,3)
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 4 (continued) Mode
Frequency (Hz) Discrepancy (%)
EMA
E10 vs. F10
1050 vs. 1075 2.4
0.92 (5,2)
E11 vs. F11
1220 vs. 1192 2.3
0.72 (4,3)
E12 vs. F12
1300 vs. 1278 1.7
0.75 (1,4)
E13 vs. F13
1400 vs. 1349 3.6
0.68 (2,4)
F14
1429 N/A
E15 vs. F15
1590 vs. 1596 0.4
0.93 (6,1)
E16 vs. F16
1710 vs. 1739 1.7
0.78 (5,3)
E17 vs. F17
1740 vs. 1759 1.1
0.84 (3,4)
E18 vs. F18
1980 vs. 1974 0.3
0.91 (6,2)
N/A
FEA
MAC (m,n)
N/A (4,4)
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Fig. 7. Representative FRFs and corresponding coherence functions for packaged PCB with fixed boundary condition.
material properties as well as the spring constant of the packaged PCB. For the free boundary condition, design variables include Young’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 order of Young’s modulus, shear modulus, Poisson’s ratio, and mass density with respect to the minimal sum of square errors between numerical and experimental natural frequencies and the criterion was set at a relative error within 10%. The optimization followed the sub-problem approximation method provided by the ANSYS software package (ANSYS, Inc., Canonsburg, PA, USA) with random design generation and sweep generation. After the optimal material properties were obtained for the packaged PCB using results corresponding to the free boundary condition, the spring constant was then solely determined following the same optimization process based on numerical and experimental natural frequencies corresponding to the fixed boundary condition. Material properties of individual components can certainly be measured independently. However, since in this work the components were modeled in a compact manner that neglected all the small constituents in order to reduce the problem size, the optimized overall material properties of individual components should more or less involve structural interactions between the components. As a result, the optimized material properties of individual components should be regarded as parameters corresponding to the specific modeling in this work, and may not be directly comparable with those from independent measurements. 3. Results and discussion 3.1. Free boundary condition For the free boundary condition, we measured hammering force and acceleration responses corresponding to 493 hammering loca-
tions on the surface of the packaged PCB. Hammering was performed three times at each of the 493 locations. Experimental FRFs corresponding to individual hammering locations were curve-fitted using ME’scope VES v. 3.0 (Vibrant Technology, Inc., Scotts Valley, CA, USA) to obtain experimental modal parameters of the packaged PCB, whose material properties were then optimized with respect to natural frequencies and listed in Table 3. The measure of anisotropy of the PCB is 2Gb(1 + mb)/ Eb = 82.8% with the optimized material properties. Our study also showed that no eligible correlation could be obtained in the present case when the PCB was assumed to be isotropic. We denote a pair (i, j) in which i stands for the output signal location, i.e., the accelerometer location and j stands for the input signal 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 corresponding coherence functions for (465,180) and (465,465), for which the latter indicates that the hammering location is right beside the accelerometer. In the figure, the synthesized curve refers to the FRF after curve-fitting based on the multiple-mode method to compensate limited signal resolution so that modal parameters can be properly estimated; the theoretical curve represents the FE solution based on the optimized material properties and measured damping of the packaged PCB, which is summarized in Section 3.3. From the figure we note that measured and numerical FRFs are in reasonably good agreement. Moreover, the coherence functions approach unity up to 2 kHz except at the frequencies where anti-resonance [17] occurs, indicating that at each hammering location, the three hammerings generated consistent results and hence the experiments were reliable. To further verify the equivalent FE model with the optimized material properties, we compare its natural frequencies and corresponding mode shapes up to 2 kHz with EMA results using the mode assurance criterion (MAC) [17], as shown in Table 4. In the table, En and Fn refer to EMA and FEA results, respectively, for the
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 5 Natural frequencies and corresponding mode shapes for packaged PCB with fixed boundary condition Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
MAC (m,n)
E1 vs. F1
194 vs. 192 1.3
0.99 (1,1)
E2 vs. F2
364 vs. 380 4.3
0.99 (1,2)
E3 vs. F3
487 vs. 491 0.9
0.96 (2,1)
E4 vs. F4
549 vs. 553 0.7
0.95 (3,3)
E5 vs. F5
695 vs. 696 0.1
0.94 (4,1)
E6 vs. F6
833 vs. 850 2.0
0.66 (4,2)
E7 vs. F7
845 vs. 860 1.7
0.84 (3,3)
E8 vs. F8
1100 vs. 1100 0.0
0.93 (5,1)
(continued on next page)
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Table 5 (continued) Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
E9 vs. F9
1290 vs. 1290 0.0
0.80 (4,3)
E10 vs. F10
1320 vs. 1311 0.7
0.89 (5,4)
E11 vs. F11
1500 vs. 1448 3.5
0.45 (5,4)
F12
1517 N/A
E13 vs. F13
1670 vs. 1699 1.7
0.40 (5,4)
E14 vs. F14
1750 vs. 1757 0.4
0.58 (4,4)
E15 vs. F15
1880 vs. 1874 0.3
0.46 (5,2)
N/A
N/A (3,2)
n-th mode. Modal characteristics along longitudinal and lateral directions are shown in parenthesis. The MAC, defined by
MACðf/x g; f/p gÞ ¼
jf/x gT f/p gj2 f/x gT f/x gf/p gT f/p g
;
MAC (m,n)
ð1Þ
is frequently used to quantitatively evaluate the discrepancy between measured mode shape, {/x}, and predicted mode shape,
{/p}. The asterisk in Eq. (1) denotes the conjugate. For two coincident mode shapes, MAC = 1. It is clear from Table 4 that there are 18 modes up to 2 kHz for the packaged PCB with the free boundary condition. The EMA result for the 14th mode could not be obtained because the accelerometer may locate at a nodal point of this particular mode. For remaining modes, natural frequencies obtained by EMA and FEA
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 6 Modal damping ratios of packaged PCB Mode
Free boundary
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17
Fixed boundary
Experimental (%)
Accumulated average (%)
Experimental (%)
Accumulated average (%)
2.139 0.577 1.300 0.541 0.475 0.675 0.620 0.840 0.549 0.722 0.675 0.518 0.563 0.760 0.577 0.601 1.251
2.139 0.939 1.339 1.139 1.006 0.951 0.904 0.896 0.857 0.844 0.828 0.803 0.784 0.782 0.769 0.758 0.787
1.432 0.698 1.693 1.658 0.777 0.772 3.153 0.574 0.707 0.888 0.865 0.668 0.759 0.928
1.432 1.065 1.274 1.370 1.252 1.172 1.455 1.345 1.274 1.235 1.202 1.157 1.126 1.112
are in good agreement with discrepancies within 7%. Mode shapes from EMA and FEA also agree very well with each other with MACs greater than 0.9 except for only a few modes whose MACs are small. The MACs are particularly small for the 4th and 5th modes, whose natural frequencies are nearly indistinguishable at around 620 Hz for the bare packaged PCB with no accelerometer attached. Clearly, for these two close modes, the simplification of the accelerometer by mass elements was only able to capture their mode shapes qualitatively. 3.2. Fixed boundary condition Following the same grid numbering as in the case of the free boundary condition, the accelerometer in the case of the fixed boundary condition located close to the 468th point. Representative FRFs for (468,74) and (468,358) up to 2 kHz are shown in Fig. 7. Clearly, measured and numerical FRFs are in reasonably good agreement while the coherence functions approach the unity except at the frequencies where anti-resonance occurs. With the optimized material properties, the spring constant for the packaged PCB with the fixed boundary condition was determined as 2.78 MN/m through the optimization process. Table 5 summarizes and compares natural frequencies and corresponding mode shapes up to 2 kHz obtained by EMA and FEA. The comparison of mode shapes is shown along lines A1 through A4 because hammering was performed along these lines only to speed up the experiment. It is clear there are 15 modes up to 2 kHz for the packaged PCB with the fixed boundary condition. The EMA result for the 12th mode could not be obtained because the accelerometer may locate at a nodal point of this particular mode. For remaining modes, natural frequencies obtained by EMA and FEA are in very good agreement with discrepancies less than 2%. Mode shapes from EMA and FEA also agree very well with each other with MACs greater than 0.9 except for only a few modes whose MACs are small. 3.3. Modal damping ratios Modal damping ratios can only be obtained from experimental measurements. In Table 6 we show modal and accumulated average damping ratios for the packaged PCB with free and fixed boundary conditions. The accumulated average damping ratio is defined by
n ¼
Pn
i¼1 ni
n
;
ð2Þ
where ni represents the damping ratio of the i-th mode. Since most of the software packages do not allow the input of modal damping ratios, the accumulated average damping ratio stands for an essential structural system characteristic in practice. From the table we note that in general a higher mode has a smaller damping ratio. These damping ratios are beneficial to the transient analysis of the packaged PCB that follows the concept of modal superposition [23,24], although this linear superposition concept can only be regarded as a qualitative estimation under severe dynamic loading conditions that bring in significant structural nonlinearity in the responses of the packaged PCB. 4. Conclusions An equivalent FE model for a standard 132 77 1 mm JEDEC drop test PCB mounted with 13 13 mm packages in a 3 5 full array has been established based on EMA results corresponding to free and fixed boundary conditions. Natural frequencies and mode shapes obtained by EMA and FEA are generally in very good agreement. We also provide modal damping ratios obtained experimentally, which would benefit the transient analysis of the packaged PCB that follows the concept of modal superposition. As a concluding remark, we have demonstrated in this work that overall material properties and screwing tightness of a packaged PCB can be determined by incorporating EMA and FEA along with an optimization process. This technique is conducive to the FE modeling of complex PCB structures for the design for mechanical reliability. References [1] Wong EH, Rajoo R, Mai Y-W, Seah SKW, Tsai KT, Yap LM. Drop impact: fundamentals and impact characterisation of solder joints. In: Proceedings of the 55th electron components and technology conference, Lake Buena Vista, FL, USA; 2005. p. 1202–9. [2] Lai YS, Chang HC, Yeh CL. Evaluation of solder joint strengths under ball impact test. Microelectron Reliab 2007;47(12):2179–87. [3] Mercado LL, Phillips B, Sahasrabudhe S, Sedillo JP, Bray D, Monroe E, et al. Handheld use condition-based bend test development. IEEE Trans Adv Packag 2006;29(2):240–9. [4] Lai Y-S, Wang TH, Tsai H-H, Jen M-HR. Cyclic bending reliability of wafer-level chip-scale packages. Microelectron Reliab 2007;47(1):111–7. [5] Lai YS, Yang PF, Yeh C-L. Experimental studies of board-level reliability of chipscale packages subjected to JEDEC drop test condition. Microelectron Reliab 2006;46(2–4):645–50. [6] Chong DYR, Che FX, Pang JHL, Ng K, Tan JYN, Low PTH. Drop impact reliability testing for lead-free and lead-based soldered IC packages. Microelectron Reliab 2006;46(7):1160–71. [7] Yang QJ, Pang HLJ, Wang ZP, Lim GH, Yap FF, Lin RM. Vibration reliability characterization of PBGA assemblies. Microelectron Reliab 2000;40(7):1097–107.
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[8] Wang H, Zhao M, Guo Q. Vibration fatigue experiments of SMT solder joint. Microelectron Reliab 2004;44(7):1143–56. [9] Kim Y, Noguchi H, Amagai A. Vibration fatigue reliability of BGA-IC package with Pb-free solder and Pb–Sn solder. Microelectron Reliab 2006;46(2– 4):459–66. [10] Liu X, Sooklal VK, Verges MA, Larson MC. Experimental study and life prediction on high cycle vibration fatigue in BGA packages. Microelectron Reliab 2006;46(7):1128–38. [11] Gu J, Barker D, Pecht M. Prognostics implementation of electronics under vibration loading. Microelectron Reliab 2007;47(12):1849–56. [12] Chen YS, Wang CS, Yang YJ. Combining vibration test with finite element analysis for the fatigue life estimation of PBGA components. Microelectron Reliab 2008;48(4):638–44. [13] JEDEC Solid State Technology Association, JESD22-B110: Subassembly Mechanical Shock; 2001. [14] JEDEC Solid State Technology Association, JESD22-B103-B: Vibration, Variable Frequency; 2002. [15] JEDEC Solid State Technology Association, JESD22-B111: Board Level Drop Test Method of Component for Handheld Electronics Products; 2003. [16] JEDEC Solid State Technology Association, JESD22-B113: Board Level Cyclic Bend Test Method for Interconnect Reliability Characterization of Components for Handheld Electronic Products; 2006.
[17] Ewins DJ. Modal Testing: Theory, practice and application, 2nd ed., Research Studies Press, Baldock, Hertfordshire, UK; 2000. [18] Yang QJ, Lim GH, Lin RM, Yap FF, Pang HLJ, Wang ZP. Experimental modal analysis of PBGA printed circuit board assemblies. In: Proceedings of the first electronic packaging technology conference, Singapore; 1997. p. 290–6. [19] Gibson RF. Modal vibration response measurements for characterization of composite materials and structures. Compos Sci Technol 2000;60(15): 2769–80. [20] He X, Fulton R. Modeling and simulation of the dynamic response of the electronic packaging. In: Proceedings of the 50th electron components and technology conference, Las Vegas, NV, USA; 2000. p. 1535–47. [21] Pitarresi J, Geng P, Beltman W, Ling Y. Dynamic modeling and measurement of personal computer motherboards. In: Proceedings of the 52nd electronic components and technology conference, San Diego, CA, USA; 2002. p. 597–603. [22] Yeh CL, Lai Y-S. Support excitation scheme for transient analysis of JEDEC board-level drop test. Microelectron Reliab 2006;46(2-4):626–36. [23] Tsai TY, Yeh CL, Lai Y-S, Chen R-S. Response spectra analysis for undamped structural systems subjected to half-sine impact acceleration pulses. Microelectron Reliab 2007;47(8):1239–45. [24] Yeh CL, Tsai TY, Lai Y-S. Transient analysis of drop responses of electronic packages using response spectra incorporated with modal superposition. Microelectron Reliab 2007;47(12):2188–96.
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Finite element model verification for packaged printed circuit board by 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, Taiwan Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan c Department of Mechanical Engineering, National Pingtung University of Science and Technology, Pingtung 912, Taiwan d Shanghai Engineering Center, Advanced Semiconductor Engineering, Inc., Shanghai 201203, China b
a r t i c l e
i n f o
Article history: Received 28 March 2008 Received in revised form 22 July 2008 Available online 12 September 2008
a b s t r a c t In this work, the experimental modal analysis (EMA) was performed to establish an equivalent finite element (FE) model for a standard Joint Electron Device Engineering Council (JEDEC) drop test printed circuit board (PCB) mounted with packages in a full array. Material properties of the equivalent FE model of the packaged PCB were calibrated through an optimization process with respect to natural frequencies based on EMA results obtained with a free boundary condition. The model was then applied to determine screwing tightness of the packaged PCB corresponding to a fixed boundary condition with the four corners of the PCB constrained, as defined by JEDEC for a board-level drop test. Modal damping ratios of the packaged PCB were also provided. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The integrity of solder joints in electronic packages under medium to harsh dynamic loading environments has become a critical issue with the prevalence of portable electronic devices as well as the introduction of stiff and brittle Pb-free solder alloys [1,2]. Several accelerated board-level reliability testing standards aiming at evaluating solder joint reliability that corresponds to these dynamic loading environments, such as cyclic bend [3,4], drop impact [5,6], or vibration [7–12], have been proposed by the Joint Electron Device Engineering Council (JEDEC) [13–16] and have been followed by the industry in qualifying the products. For the design purpose, the finite element analysis (FEA) has long been proven to be conducive in selecting proper structural configurations and materials for electronic packages without the need of costly and time consuming experiments. However, the accuracy of numerical solutions depends greatly on the feasibility of modeling that includes proper settings of geometry, boundary and loading conditions, and material properties. For FEA of a board-level test, modeling of the printed circuit board (PCB) is generally considered as the source that brings the most uncertainty to the numerical solutions. Overall mechanical properties of the PCB can vary according to different numbers of metal layers, different layouts of circuits, and different polymeric and composite reinforcing materials used in the fabrication. Tiny circuits and their complex layouts inside the PCB also limit the possibility of comprehensive modeling of the PCB.
* Corresponding author. Tel.: +886 6 2757575x63328; fax: +886 6 2766549. E-mail address: [email protected] (R.-S. Chen). 0026-2714/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2008.07.068
We note, however, if the concern is only with overall mechanical properties of the PCB or the board- or system-level test vehicle that contains PCB and packages mounted onto it, an experimental modal analysis (EMA) [17] can practically determine modal parameters, including natural frequencies, mode shapes, and damping ratios, of the specific PCB or board- or system-level test vehicle without the need of prerequisite information of material properties and layouts of its individual constituent components [18–20]. An equivalent finite element (FE) model calibrated by modal parameters obtained from EMA can thus be used with more flexibility for the design purpose [5,21]. In the present work, we followed EMA to characterize modal parameters 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 fixed boundary conditions are shown in Fig. 2. As depicted by the procedure in Fig. 3, material properties of the equivalent FE model of the packaged PCB were calibrated through an optimization process based on EMA results obtained with a free boundary condition. The model was then applied to determine screwing tightness of the packaged PCB corresponding to a fixed boundary condition with the four corners of the PCB constrained, as defined by JEDEC for a board-level drop test [13,15].
2. EMA and FEA Experimental setups for EMA on the packaged PCB with free and fixed boundary conditions have been shown in Fig. 2. To avoid interference, hammering was performed on the reversed side of
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846
Fig. 1. Schematic of packaged PCB (not to the scale).
the packaged surface, Fig. 4, in which the locations where the accelerometer was affixed are labeled. For the free boundary condition, we measured hammering force and acceleration responses corresponding to hammering locations in a 17 (vertical) 29 (horizontal) array on the surface, shown by the black dots in Fig. 4. For the fixed boundary condition, hammering force and acceleration responses corresponding to 88 hammering locations along the A1 through A4 lines shown in Fig. 4 were measured. The frequency response functions (FRFs) between measured hammering force and acceleration responses were then curve-fitted to extract experimental modal parameters of the packaged PCB. Fig. 5 shows full three-dimensional FE models built for the packaged PCB with free and fixed boundary conditions. The model for the packaged PCB contained 12,335 linear hexahedral solid elements along with 24 mass elements to take into account the accelerometer mass. For the fixed boundary condition, 192 additional linear spring-damper elements were employed to account for the tightness of screws at the four corners; the torque applied to tighten the screws was 8 kgf-m. The damping effect of the screws was neglected in this study while only the spring effect was considered. The spring constant was assumed to be constant and was to be determined in the subsequent optimization procedure after the material properties of the packaged PCB were determined from the optimization based on EMA and FEA results for the free boundary condition. Initial material properties extracted from Yeh and Lai [22], listed in Table 1, were specified to PCB and packages. We assume that the packages are isotropic while the PCB is cubic with three
Fig. 3. Procedure for equivalent FE model establishment.
Fig. 4. Grid of measurement points (Accelerometer locations: N for free; j for fixed).
Fig. 2. Packaged PCB with free and fixed boundary conditions (packages on reversed side).
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Fig. 5. FE models of packaged PCB.
Table 3 Comparison between initial and optimized material properties of packaged PCB
Table 1 Initial material properties of packaged PCB [22]
Young’s modulus (GPa) Poisson’s ratio Mass density (g/cm3)
PCB
Package
15.50 0.20 1.91
28.00 0.35 1.81
Eb (GPa) Gb (GPa)
mb qb (g/cm3) Ep (GPa)
mp qp (g/cm3)
Table 2 Convergence analysis with respect to natural frequencies (Unit: Hz) Cell
A
B
C
D
B vs. C
Relative error (%)
Number of elements
Mode 1 2 3 4 5 6
3620
7245
12,335
17,415
178.97 220.50 465.73 491.95 495.89 772.80
148.91 191.13 393.58 394.27 401.51 562.22
147.34 189.46 386.87 391.00 395.20 551.21
145.48 188.12 385.85 388.19 388.39 549.91
1.054 0.875 1.705 0.829 1.572 1.958
Initial
Optimized
Discrepancy (%)
15.50 6.46 0.20 1.91 28.00 0.35 1.81
9.42 3.12 0.25 2.05 20.00 0.4 1.84
39.2 51.6 25.0 6.8 28.6 14.3 1.6
independent elastic constants, albeit isotropy was given to the PCB as an initial setting. The accelerometer mass is 1.5 g. A numerical convergence analysis based on the initial material properties was carried out with respect to natural frequencies of the first five modes of the packaged PCB, as shown in Table 2. Clearly from the results, the FE model with 12,335 elements has achieved sufficient accuracy with moderate computational efficiency. An optimization process based on the coincidence of numerical and experimental natural frequencies was followed to iterate
Fig. 6. Representative FRFs and corresponding coherence functions for packaged PCB with free boundary condition.
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Table 4 Natural frequencies and corresponding mode shapes for packaged PCB with free boundary condition Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
MAC (m,n)
E1 vs. F1
161 vs. 154 4.3
0.91 (3,1)
E2 vs. F2
187 vs. 200 7.0
0.83 (3,1)
E3 vs. F3
356 vs. 357 0.3
0.91 (3,2)
E4 vs. F4
461 vs. 457 0.9
0.68 (4,1)
E5 vs. F5
492 vs. 467 5.1
0.70 (1,3)
E6 vs. F6
582 vs. 574 1.4
0.97 (2,3)
E7 vs. F7
660 vs. 684 3.6
0.97 (4,2)
E8 vs. F8
858 vs. 857 1.2
0.97 (5,1)
E9 vs. F9
913 vs. 921 0.9
0.94 (3,3)
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 4 (continued) Mode
Frequency (Hz) Discrepancy (%)
EMA
E10 vs. F10
1050 vs. 1075 2.4
0.92 (5,2)
E11 vs. F11
1220 vs. 1192 2.3
0.72 (4,3)
E12 vs. F12
1300 vs. 1278 1.7
0.75 (1,4)
E13 vs. F13
1400 vs. 1349 3.6
0.68 (2,4)
F14
1429 N/A
E15 vs. F15
1590 vs. 1596 0.4
0.93 (6,1)
E16 vs. F16
1710 vs. 1739 1.7
0.78 (5,3)
E17 vs. F17
1740 vs. 1759 1.1
0.84 (3,4)
E18 vs. F18
1980 vs. 1974 0.3
0.91 (6,2)
N/A
FEA
MAC (m,n)
N/A (4,4)
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Fig. 7. Representative FRFs and corresponding coherence functions for packaged PCB with fixed boundary condition.
material properties as well as the spring constant of the packaged PCB. For the free boundary condition, design variables include Young’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 order of Young’s modulus, shear modulus, Poisson’s ratio, and mass density with respect to the minimal sum of square errors between numerical and experimental natural frequencies and the criterion was set at a relative error within 10%. The optimization followed the sub-problem approximation method provided by the ANSYS software package (ANSYS, Inc., Canonsburg, PA, USA) with random design generation and sweep generation. After the optimal material properties were obtained for the packaged PCB using results corresponding to the free boundary condition, the spring constant was then solely determined following the same optimization process based on numerical and experimental natural frequencies corresponding to the fixed boundary condition. Material properties of individual components can certainly be measured independently. However, since in this work the components were modeled in a compact manner that neglected all the small constituents in order to reduce the problem size, the optimized overall material properties of individual components should more or less involve structural interactions between the components. As a result, the optimized material properties of individual components should be regarded as parameters corresponding to the specific modeling in this work, and may not be directly comparable with those from independent measurements. 3. Results and discussion 3.1. Free boundary condition For the free boundary condition, we measured hammering force and acceleration responses corresponding to 493 hammering loca-
tions on the surface of the packaged PCB. Hammering was performed three times at each of the 493 locations. Experimental FRFs corresponding to individual hammering locations were curve-fitted using ME’scope VES v. 3.0 (Vibrant Technology, Inc., Scotts Valley, CA, USA) to obtain experimental modal parameters of the packaged PCB, whose material properties were then optimized with respect to natural frequencies and listed in Table 3. The measure of anisotropy of the PCB is 2Gb(1 + mb)/ Eb = 82.8% with the optimized material properties. Our study also showed that no eligible correlation could be obtained in the present case when the PCB was assumed to be isotropic. We denote a pair (i, j) in which i stands for the output signal location, i.e., the accelerometer location and j stands for the input signal 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 corresponding coherence functions for (465,180) and (465,465), for which the latter indicates that the hammering location is right beside the accelerometer. In the figure, the synthesized curve refers to the FRF after curve-fitting based on the multiple-mode method to compensate limited signal resolution so that modal parameters can be properly estimated; the theoretical curve represents the FE solution based on the optimized material properties and measured damping of the packaged PCB, which is summarized in Section 3.3. From the figure we note that measured and numerical FRFs are in reasonably good agreement. Moreover, the coherence functions approach unity up to 2 kHz except at the frequencies where anti-resonance [17] occurs, indicating that at each hammering location, the three hammerings generated consistent results and hence the experiments were reliable. To further verify the equivalent FE model with the optimized material properties, we compare its natural frequencies and corresponding mode shapes up to 2 kHz with EMA results using the mode assurance criterion (MAC) [17], as shown in Table 4. In the table, En and Fn refer to EMA and FEA results, respectively, for the
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 5 Natural frequencies and corresponding mode shapes for packaged PCB with fixed boundary condition Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
MAC (m,n)
E1 vs. F1
194 vs. 192 1.3
0.99 (1,1)
E2 vs. F2
364 vs. 380 4.3
0.99 (1,2)
E3 vs. F3
487 vs. 491 0.9
0.96 (2,1)
E4 vs. F4
549 vs. 553 0.7
0.95 (3,3)
E5 vs. F5
695 vs. 696 0.1
0.94 (4,1)
E6 vs. F6
833 vs. 850 2.0
0.66 (4,2)
E7 vs. F7
845 vs. 860 1.7
0.84 (3,3)
E8 vs. F8
1100 vs. 1100 0.0
0.93 (5,1)
(continued on next page)
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Table 5 (continued) Mode
Frequency (Hz) Discrepancy (%)
EMA
FEA
E9 vs. F9
1290 vs. 1290 0.0
0.80 (4,3)
E10 vs. F10
1320 vs. 1311 0.7
0.89 (5,4)
E11 vs. F11
1500 vs. 1448 3.5
0.45 (5,4)
F12
1517 N/A
E13 vs. F13
1670 vs. 1699 1.7
0.40 (5,4)
E14 vs. F14
1750 vs. 1757 0.4
0.58 (4,4)
E15 vs. F15
1880 vs. 1874 0.3
0.46 (5,2)
N/A
N/A (3,2)
n-th mode. Modal characteristics along longitudinal and lateral directions are shown in parenthesis. The MAC, defined by
MACðf/x g; f/p gÞ ¼
jf/x gT f/p gj2 f/x gT f/x gf/p gT f/p g
;
MAC (m,n)
ð1Þ
is frequently used to quantitatively evaluate the discrepancy between measured mode shape, {/x}, and predicted mode shape,
{/p}. The asterisk in Eq. (1) denotes the conjugate. For two coincident mode shapes, MAC = 1. It is clear from Table 4 that there are 18 modes up to 2 kHz for the packaged PCB with the free boundary condition. The EMA result for the 14th mode could not be obtained because the accelerometer may locate at a nodal point of this particular mode. For remaining modes, natural frequencies obtained by EMA and FEA
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Y.-C. Lee et al. / Microelectronics Reliability 48 (2008) 1837–1846 Table 6 Modal damping ratios of packaged PCB Mode
Free boundary
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17
Fixed boundary
Experimental (%)
Accumulated average (%)
Experimental (%)
Accumulated average (%)
2.139 0.577 1.300 0.541 0.475 0.675 0.620 0.840 0.549 0.722 0.675 0.518 0.563 0.760 0.577 0.601 1.251
2.139 0.939 1.339 1.139 1.006 0.951 0.904 0.896 0.857 0.844 0.828 0.803 0.784 0.782 0.769 0.758 0.787
1.432 0.698 1.693 1.658 0.777 0.772 3.153 0.574 0.707 0.888 0.865 0.668 0.759 0.928
1.432 1.065 1.274 1.370 1.252 1.172 1.455 1.345 1.274 1.235 1.202 1.157 1.126 1.112
are in good agreement with discrepancies within 7%. Mode shapes from EMA and FEA also agree very well with each other with MACs greater than 0.9 except for only a few modes whose MACs are small. The MACs are particularly small for the 4th and 5th modes, whose natural frequencies are nearly indistinguishable at around 620 Hz for the bare packaged PCB with no accelerometer attached. Clearly, for these two close modes, the simplification of the accelerometer by mass elements was only able to capture their mode shapes qualitatively. 3.2. Fixed boundary condition Following the same grid numbering as in the case of the free boundary condition, the accelerometer in the case of the fixed boundary condition located close to the 468th point. Representative FRFs for (468,74) and (468,358) up to 2 kHz are shown in Fig. 7. Clearly, measured and numerical FRFs are in reasonably good agreement while the coherence functions approach the unity except at the frequencies where anti-resonance occurs. With the optimized material properties, the spring constant for the packaged PCB with the fixed boundary condition was determined as 2.78 MN/m through the optimization process. Table 5 summarizes and compares natural frequencies and corresponding mode shapes up to 2 kHz obtained by EMA and FEA. The comparison of mode shapes is shown along lines A1 through A4 because hammering was performed along these lines only to speed up the experiment. It is clear there are 15 modes up to 2 kHz for the packaged PCB with the fixed boundary condition. The EMA result for the 12th mode could not be obtained because the accelerometer may locate at a nodal point of this particular mode. For remaining modes, natural frequencies obtained by EMA and FEA are in very good agreement with discrepancies less than 2%. Mode shapes from EMA and FEA also agree very well with each other with MACs greater than 0.9 except for only a few modes whose MACs are small. 3.3. Modal damping ratios Modal damping ratios can only be obtained from experimental measurements. In Table 6 we show modal and accumulated average damping ratios for the packaged PCB with free and fixed boundary conditions. The accumulated average damping ratio is defined by
n ¼
Pn
i¼1 ni
n
;
ð2Þ
where ni represents the damping ratio of the i-th mode. Since most of the software packages do not allow the input of modal damping ratios, the accumulated average damping ratio stands for an essential structural system characteristic in practice. From the table we note that in general a higher mode has a smaller damping ratio. These damping ratios are beneficial to the transient analysis of the packaged PCB that follows the concept of modal superposition [23,24], although this linear superposition concept can only be regarded as a qualitative estimation under severe dynamic loading conditions that bring in significant structural nonlinearity in the responses of the packaged PCB. 4. Conclusions An equivalent FE model for a standard 132 77 1 mm JEDEC drop test PCB mounted with 13 13 mm packages in a 3 5 full array has been established based on EMA results corresponding to free and fixed boundary conditions. Natural frequencies and mode shapes obtained by EMA and FEA are generally in very good agreement. We also provide modal damping ratios obtained experimentally, which would benefit the transient analysis of the packaged PCB that follows the concept of modal superposition. As a concluding remark, we have demonstrated in this work that overall material properties and screwing tightness of a packaged PCB can be determined by incorporating EMA and FEA along with an optimization process. This technique is conducive to the FE modeling of complex PCB structures for the design for mechanical reliability. References [1] Wong EH, Rajoo R, Mai Y-W, Seah SKW, Tsai KT, Yap LM. Drop impact: fundamentals and impact characterisation of solder joints. In: Proceedings of the 55th electron components and technology conference, Lake Buena Vista, FL, USA; 2005. p. 1202–9. [2] Lai YS, Chang HC, Yeh CL. Evaluation of solder joint strengths under ball impact test. Microelectron Reliab 2007;47(12):2179–87. [3] Mercado LL, Phillips B, Sahasrabudhe S, Sedillo JP, Bray D, Monroe E, et al. Handheld use condition-based bend test development. IEEE Trans Adv Packag 2006;29(2):240–9. [4] Lai Y-S, Wang TH, Tsai H-H, Jen M-HR. Cyclic bending reliability of wafer-level chip-scale packages. Microelectron Reliab 2007;47(1):111–7. [5] Lai YS, Yang PF, Yeh C-L. Experimental studies of board-level reliability of chipscale packages subjected to JEDEC drop test condition. Microelectron Reliab 2006;46(2–4):645–50. [6] Chong DYR, Che FX, Pang JHL, Ng K, Tan JYN, Low PTH. Drop impact reliability testing for lead-free and lead-based soldered IC packages. Microelectron Reliab 2006;46(7):1160–71. [7] Yang QJ, Pang HLJ, Wang ZP, Lim GH, Yap FF, Lin RM. Vibration reliability characterization of PBGA assemblies. Microelectron Reliab 2000;40(7):1097–107.
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