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Simplified modelling of printed circuit boards for spacecraft applications
Acta Astronautica 65 (2009) 192 – 201 www.elsevier.com/locate/actaastro
Simplified modelling of printed circuit boards for spacecraft applications Robin Alastair Amya,∗ , G.S. Agliettia , Guy Richardsonb a Astronautical Research Group, University of Southampton, School of Engineering Sciences, Southampton, UK b Surrey Satellite Technology Limited, Guilford, Surrey, UK
Received 22 December 2006; accepted 13 January 2009 Available online 24 February 2009
Abstract The severe mechanical environment that electronic components experience during spacecraft launch necessitates that their failure probability be assessed. One possible approach is to create an accurate model of the printed circuit board’s (PCBs) dynamic response; subsequently, the failure probability can be determined by comparing the response model with corresponding failure criteria for the electronic components. In principle the response model can be achieved by a very detailed finite element (FE) model of the PCB which would include the mass and stiffness of all components present on the PCB. Unfortunately this approach requires an excessive effort; therefore, it is rarely pursued by the designer. Past research has shown that assumptions can be made about the mass and stiffness that allow simpler models to be created that still achieve appropriate levels of accuracy. However, the accuracy of these simplified models has not yet been quantified over a range of possible design cases. This paper will quantify how increasing levels of modelling simplifications decrease the accuracy of PCB FE models. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction During the launch phase, spacecraft electronic equipment is subjected to a very harsh vibration environment. If the equipment is not designed to withstand this vibration, the components soldered to the printed circuit board (PCB) may fail. The current process used to ensure that failure does not occur is a trial and error approach mixed with a certain amount of subjective experience, this approach necessitates that a prototype of the board be manufactured and subjected to a representative environment. Generally this approach is cost effective as most prototypes do not fail the vibration ∗ Corresponding author.
E-mail address: [email protected] (R.A. Amy). 0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.01.037
qualification test, but if a prototype does fail the resultant iterative trial and error process can become very costly and time consuming. One solution is to analytically predict the failure by the following two-stage physics-of-failure (PoF) approach. The first stage calculates the vibration response of the board through a finite element (FE) model of the PCB/component system, incorporating some simplifications and assumptions to speed up the modelling process. The second stage relates this calculated response to some pre-determined component failure criteria, to show whether or not the component can withstand the predicted response (e.g. curvature or acceleration). The principal short-coming of the first stage of the PoF process is that the accuracy of the process is unknown, thus the confidence in the failure criteria and consequent reliability estimates cannot be assessed.
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This work will illustrate a process to determine this error dependant on the simplification assumptions used. The process is a Monte Carlo simulation of randomly created PCBs at various levels of simplification, with each run being compared to an unsimplified benchmark case. It will then be possible to see the error statistics for each level of simplification. In addition to examining how the relative error varies within each level of simplification, it is also useful to find the correlation of relative error with the geometry and mechanical properties (input variables) of the PCB/component systems, for example, it might be possible to accept a higher level of simplification for smaller or less heavily populated boards. The main practical uses of this process are either to determine an appropriate level of model simplification before a modelling commences or to determine the expected level of error for a pre-existing model. In the first case an allowable error would be given, along with the geometry of the proposed PCB, from this data a simplification regime could be chosen. In the second case a model would be given, along with the levels of simplification used in its creation, from this data the expected error could be calculated. In addition to these two uses it is also possible to gain insight into which geometry and mechanical properties correlate with high stress on components, this knowledge can be easily used to increase the reliability. Essentially the process is a sensitivity analysis of FE models of PCBs, with the inputs being the geometry and mechanical properties. The output is the relative error of the deformed shape with respect to a benchmark case, the relative error is not an absolute value, only the ratio of the simplified system to the benchmark case. Because of these assumptions no experimental analysis is considered necessary, as it would only be measuring the accuracy of the benchmark case, which in this case is not of importance. What is of importance here is the ratio of the results due to the simplification and input variables. 2. Current state of the art Attaching components to a PCB will make the underlying PCB seem to behave as if it has increased density and stiffness properties. If this increase is known for each component, an FE model can be made with the increased density and stiffness properties at each expected component location. To simplify this approach the usual process is to sum the additional mass and stiffness increases provided by the attached components and “smear” this averaged property over the entire FE model
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of the PCB. This “smearing” process has been previously validated for some specific cases of electronic equipment [1,2], where the validation criteria was error in natural frequencies and the model assurance criterion (MAC). Another area of simplification is in the specification of the model’s edge rotational stiffness; due to lack of information this value is usually modelled as zero (i.e. simply supported). It has been noted that this simplification reduces the accuracy in predicting the overall response much more significantly than property smearing, especially in cases where the edges are firmly clamped [3,4]. Once the response of the PCB to vibration input has been calculated it must be compared to some preexisting failure criteria. Early work focused on the overall RMS acceleration input as the primary cause of failure [5,6]; however, it has been recognised that the individual component local acceleration environment is much more important in contributing to failure [6]. From similar observations, other research has tried to improve the state of the art by considering failure due to the local acceleration environment experienced by each individual component [7]. Recent work has considered the local PCB curvature as the failure criteria as opposed to the local acceleration. This approach is especially pertinent for surface mount technology (SMT) components [6,8–10]. This work improves upon the current state of the art by considering the relative error in curvatures and displacements, as opposed to previous research which only considered the MAC. MAC is deficient as it is only a measure of the overall shape difference of eigenvectors, not error of local curvature. The use of curvature as an error variable shows that although the overall shape may be very similar, small local differences around components may lead to significant error in the stress experienced by components. Furthermore the results of this work are valid over a larger range of possible cases than previous research; this is achieved by using a Monte Carlo approach with many different combinations of geometry and component locations. Finally the effect of error in the rotational edge stiffness is also examined; this focuses on the error induced by modelling the edge rotational stiffness as zero. 3. Random creation of models The Monte Carlo simulation included several hundred individual runs, for each of these runs the following input variables are used to randomly create a PCB FE model: length of first side, ratio of second edge,
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thickness of PCB and edge rotational stiffness. These variables are in various statistical distributions pertinent to the case being studied. The Young’s modulus and density of the unpopulated PCB are assumed to be constant across all models. It is now possible to populate these models with components. This is achieved by first dividing the components into three separate categories: Light components: This is intended to model small components such as resistors or transistors, the length and mass increase of the components is small, the stiffness increase is negligible. Surface mount technology: The SMT category symbolises components such as quad flat pack (QFP), ball grid array (BGA) and pin grid array (PGA), which are generally have a longest edge of about 10.30 mm, and have an increased density and stiffness ratios are in proportion to the length and inversely proportional to the thickness of the PCB to which they are attached. Heavy components: This category is intended to reflect large components such as transformers, large power capacitors and resistors. The density and stiffness ratios are proportional to component length. Each run of the Monte Carlo simulation has different weightings of the different component categories; this allows the effect of different types of components to be assessed. Using the component type weighting the FE model is randomly populated with components, with the size of each component falling within distributions appropriate to the component type. The stiffness and density increases are calculated by statistical distributions that are functions of size, component type and PCB thickness. The stiffness and density increase are achieved by modifying the local PCB Young’s modulus and density to reflect the component contributions. The penultimate step of creating the FE models is to apply the boundary conditions. The displacement degrees of freedom of the edge of the PCB are not considered, as this has been shown to be a safe assumption. The rotational degrees of freedom in the in-plane directions are set to random values between 0% and 100%, where 0% is intended to reflect a simply supported BC and 100% a clamped BC, this is based on the definitions from [4]. The final step in creating the FE models is to create the mass and stiffness matrices, this is achieved using the quad4 element type from the OpenFEM element library.
4. Levels of simplification There are two categories of simplification that are used, property smearing; where the mass or stiffness due to the addition of components is averaged over the entire PCB, and boundary condition (BC) simplification; where the rotational stiffness of the edge of the PCB is assumed or estimated. The levels of property smearing used are: Mass smearing: The area weighted density of the components are summed and averaged over the entire surface of the board. Stiffness smearing: As with mass smearing but stiffness instead. Mass and stiffness smearing: The combination of the previous two levels of smearing. Unpopulated: The simplest level of modelling assumes the PCB/component system can be modelled as the unpopulated PCB, i.e. completely ignore the mass and stiffness effects of the components. Considering the boundary condition simplification, when no other information is available it is usual to model the rotational stiffness as zero. As mentioned previously the rotational edge stiffness is given a fixity value between 0% to 100%. For more information see [3,4].
5. Overall creation and solution process It is appropriate at this point to re-iterate the whole process, which is as follows: 1. Randomly create a set of variables from pre-defined distributions; variables are edge length, edge ratio, thickness, edge stiffness. 2. From these variables create a PCB layout. 3. Randomly locate components on the FE model, where the density, probability and size of the components are within pre-defined distributions. 4. Give each component a density and stiffness ratio increase, where the ratios are dependant on the component type, size and PCB thickness. 5. Solve this initial model to create the “benchmark” case. 6. Apply the various levels of smearing to create the simplified models. 7. Solve each simplified case and compare the results with the “benchmark” case.
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8. Set the edge rotational stiffness to zero and repeat the last two steps. 9. Repeat the whole process many times (700 in the present case study). The models are created and solved in the MATLAB environment using the OpenFEM element library and solvers. The out of plane deformation is found for a unit acceleration input. The PCB curvature is subsequently calculated from the deformation at every node on the FE model, in this case study the nodes are spaced at 5 mm intervals. Comparing the percentage error at each node to its relative node on the benchmark case gives the relative curvature error, , across the model. Once this is calculated in both the x and y directions it is possible to find the statistical measures of , such as maximum, minimum, mean, interquartile range, etc. The same process is repeated for the out of plane deformation, this shall be referred to as z. The eigenvalues and eigenvectors are also calculated to examine the MAC and relative frequency error, this step is mainly for completeness and to allow further insight into the results.
6. Discussion of the results The measure of that will be used in this case study will be the 75th percentile (75th ); this is a measure of the range of the data. The data may not be centred
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around zero, to account for this the mean is added to the half the 75th percentile range. This value was used as it is robust to outliers and can also be used to approximate (standard deviation). Fig. 1 is a plot of 75th against the different levels of smearing, the first observation that can be made of this graph is that the simplification levels that model the BCs as simply supported have a much larger spread of results. This point is clarified in Fig. 2, which includes a third dimension showing the amount of edge rotational stiffness simplification that has occurred. This three dimensional plot shows that the accuracy of the results are very sensitive to errors in specifying the edge rotational stiffness, especially in those cases where the original stiffness was very high (i.e. those cases that had more to lose from simplification), causing the model to suffer a large loss in percentage edge fixity. The sensitivity of the models to the edge rotational stiffness is considerable and tends to dilute the effect of other variables, thus for the rest of this analysis it will be assumed that the edge rotational stiffness can be found and modelled precisely. This will allow the sensitivity of the modelling to other variables to be studied more easily, although it should be kept in mind that large amounts of simplification in edge stiffness will results in large errors in the model. Fig. 3 plots the first four levels of smearing (i.e. those that have no simplification of the edge rotational stiffness) against the ratio of the unpopulated PCB stiffness to the actual stiffness ratio. From this graph it can be
Fig. 1. Smearing level plotted against 75th , “both” refers to joint density and stiffness smearing, note the large variation in the last four levels due to the simplification of the boundary conditions.
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Fig. 2. Smearing level plotted against 75th and edge rotational stiffness, only the four levels of smearing that simplify BCs were plotted.
Fig. 3. Smearing level plotted against curvature delta and ratio of populated to unpopulated stiffness.
seen that there is a distinct positive correlation between the accuracy and the stiffness ratio, although the correlation is stronger for some levels of simplification than others. Fig. 4 shows this for the specific case of stiffness smearing. The correlation of is stronger with stiffness smearing than density smearing; this suggests that smearing the stiffness has a greater negative impact on curvature accuracy than smearing the density. It should be apparent from the large number of input variables, that this trial and error approach to finding the correlation of the input and output variables is a time
consuming process. To reduce the amount of time spent doing this, it is possible to create a matrix that lists the amount of correlation of all the variables against one another. In this matrix the first third of the variables were input variables (see Fig. 5), the last third were various measurements of error and the middle group represent measurements of MAC and fundamental frequency errors. If this matrix is plotted graphically with the correlation value in each cell represented as a colour, then the variables that correlate with each other immediately become obvious. Figs. 5 and 6 illustrate these
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Fig. 4. Stiffness ratio plotted against curvature delta for the stiffness smearing case.
Fig. 5. Cross-correlation of variables for the stiffness smeared case, each number is an index that relates to a specific variable, with x and y axes plotting the same variables. Note that the diagonal of the matrix shows perfect correlation, this is due to the variables being correlated against themselves. See Fig. 6 for further clarification.
correlation plots. In this case it can be quickly seen that the variables that strongly correlate with error are the PCB thickness and the stiffness and density ratios, to a lesser extent other variables also correlate with error. It is now possible to concentrate solely on the effect of
these input variables on the , the following discussion will focus on these variables. However, if no significant correlation are found between any of the input and output variables, only the global statistics (such as those in Fig. 1) can be analysed.
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Fig. 6. Magnification of lower right corner of Fig. 5, several variables have been referenced, the shade of each square represents the correlation between two variables, with darker shades indicating higher correlation.
With these observations it is possible to analyse only those values that strongly correlate with failure. Fig. 7 shows the aforementioned variables plotted together with colour representing the at each discrete point. The resulting points create a three dimensional surface, from which the expected error can be calculated. In a similar was to the previous example it is possible to plot thickness as the fourth (colour) variable, as in Fig. 8. From this it is possible to select a specific thickness (as would very likely occur during the design of a PCB), and focus on data points that relate to this specific thickness of interest whilst ignoring data points relating to other thicknesses. Fig. 9 shows this selective process, the only data points that are plotted are those for the thickness of interest (1.6 mm in this case). If the stiffness and density ratios are known, it is possible to directly assess the expected accuracy of the results. 7. Observations The following observations were noted during the case study, although these observations may not be valid in a different case that uses different input values the implications are the same. The sensitivity of the results to edge rotational stiffness: In agreement with past research [3,4] the results are very sensitive to error in specifying the edge rotational stiffness.
Some variables seem to have little correlation with error: In the cases studied here the results are insensitive to some input variables, such as ratio of edge lengths, whilst other variables show only very small correlation or the correlation depended on the level of simplification. The correlation of stiffness and density ratios with error: It was noted that the stiffness and mass ratios show a strong positive correlation with , this agrees with observations in previous research [2]. Furthermore it is noted that the out of plane displacement are sensitive to simplification of the mass properties, which is in contrast to the curvature, which is more sensitive to simplification of the stiffness properties. Thicker PCBs are less sensitive to the effect of components: Due to fact that thicker PCBs are less receptive to the additional component mass and stiffness, the accuracy of models of thick PCBs are more accurate than those of thin PCBs. In the cases studied those PCBs over 2 mm thick which only composed of light and SMT components all showed curvature error deltas within 5%.
8. Sources of modelling error The following variables were not considered in the case study illustrated, although they could be included in further studies if they were deemed significant.
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Fig. 7. Stiffness and density ratios plotted against PCB thickness, for stiffness smearing case, points are grouped by errors.
Isotropic PCB properties: Real-life PCBs are usually anisotropic due to the lay-up, in the case study the PCB properties were assumed to be isotropic as the lay-up direction might not be known. Anisotropy could be easily included in the model if deemed necessary. Homogeneous density and stiffness under components: The density and stiffness increase due to each component is assumed to be constant in the area underneath each component, further analysis may find a distributed stiffness, but it is assumed that this is a safe assumption. Homogeneous edge rotational stiffness: The rotational edge stiffness was modelled as continuous and constant for all edges, real-life situations are more likely to exhibit discrete locations of stiffness which vary from edge to edge. The homogeneous assumption was used due the excessive amount of runs that would be required to incorporate all different possible combinations of edge stiffness. It would be feasible to model this if really necessary, especially if other input variables were constant.
Zero component height: For small components and those of SMT type, this is a safe assumption, as these small components have negligible height above the PCB. Equipment with higher percentage of large tall components is more likely to be affected by this assumption, as the rocking motions associated with such components may cause stresses that are not modelled. Low frequency change: It is assumed that the change in frequencies is low, as otherwise it is possible that the modes with a significant frequency change may be actually occur at frequencies with a more intense vibration input spectrum (if the vibration input is assumed to be shaped). For the first three levels of simplification studied here, the maximum variation in frequency was typically within 5%. Whereas, if the board was modelled as unpopulated the variation could be as high as 20%, whilst for large amounts of edge stiffening simplification the variation rose to 30%. Ultimately it should be noted that for shaped vibration input, the statistics of frequency change should be taken into account when choosing a simplification regime.
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Fig. 8. Stiffness and density ratios plotted against error, for stiffness smearing case, points are grouped by thickness.
Fig. 9. Stiffness and density ratios plotted against error, for stiffness smearing case at 1.6 mm.
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9. Conclusion A process has been illustrated that allows the relative error of PCB FE models to be assessed; this is calculated at various levels of model simplification. The resulting data sets can then be analysed using the techniques suggested, eventually providing the expected relative error for given levels of smearing, which can be further decomposed into errors for various input variables. Some important observations that were made during the data analysis were that the curvature was more sensitive to simplification of the stiffness properties, whilst overall response was more sensitive to density smearing. The sensitivity of modelling accuracy to large amounts of simplification in the edge rotational stiffness was also highlighted. In addition to this, the data can also provide valuable insight into the nature of the response of a particular board, and how simple modifications to only a few variables can reduce the response with very little cost. Acknowledgement The author would like to warmly thank Surrey Satellite Technology Limited (SSTL) for their vital contribution to this research. References [1] J.M. Pitarresi, A. Primavera, Comparison of vibration modeling techniques for printed circuit cards, ASME Journal of Electronic Packaging 114 (1991) 378–383.
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[2] J.M. Pitarresi, D.V. Caletka, R. Caldwell, D.E. Smith, The “Smeared” property technique for the FE vibration analysis of printed circuit cards, ASME Journal of Electronic Packaging 113 (1991) 250–257. [3] G.H. Lim, J.H. Ong, J.E.T. Penny, Effect of edge and internal point support of a printed circuit board under vibration, ASME Journal of Electronic Packaging 121 (2) (1999) 122–126. [4] D.B. Barker, Y.S. Chen, Modeling the vibration restraints of wedge-lock card guides, ASME Journal of Electronic Packaging 115 (2) (1993) 189–194. [5] J. Lau, L.M. Powers-Maloney, J.R. Baker, D. Rice, B. Shaw, Solder joint reliability of fine pitch surface mount technology assemblies, IEEE Transactions on Components, Hybrids, and Manufacturing Technology 13 (3) (1990) 534–544. [6] R.S. Li, L. Poglitsch, Fatigue of plastic ball grid array and plastic quad flat packages under automotive vibration, in: SMTA International, Proceedings of the Technical Program, 2001, pp. 324–329. [7] S. Liguore, D. Followell, Vibration fatigue of surface mount technology (SMT) solder joints, in: Annual Reliability and Maintainability Symposium 1995 Proceedings (Cat. no. 95CH35743), 1995, pp. 18–26. [8] D.S. Steinberg, Vibration Analysis for Electronic Equipment, Wiley, 2000. [9] H. Wang, M. Zhao, Q. Guo, Vibration fatigue experiments of SMT solder joint, Microelectronics Reliability 44 (7) (2004) 1143–1156. [10] S. Shetty, V. Lehtinen, A. Dasgupta, V. Halkola, T. Reinikainen, Fatigue of chip scale package interconnects due to cyclic bending, ASME Journal of Electronic Packaging 123 (3) (2001) 302–308.
Simplified modelling of printed circuit boards for spacecraft applications Robin Alastair Amya,∗ , G.S. Agliettia , Guy Richardsonb a Astronautical Research Group, University of Southampton, School of Engineering Sciences, Southampton, UK b Surrey Satellite Technology Limited, Guilford, Surrey, UK
Received 22 December 2006; accepted 13 January 2009 Available online 24 February 2009
Abstract The severe mechanical environment that electronic components experience during spacecraft launch necessitates that their failure probability be assessed. One possible approach is to create an accurate model of the printed circuit board’s (PCBs) dynamic response; subsequently, the failure probability can be determined by comparing the response model with corresponding failure criteria for the electronic components. In principle the response model can be achieved by a very detailed finite element (FE) model of the PCB which would include the mass and stiffness of all components present on the PCB. Unfortunately this approach requires an excessive effort; therefore, it is rarely pursued by the designer. Past research has shown that assumptions can be made about the mass and stiffness that allow simpler models to be created that still achieve appropriate levels of accuracy. However, the accuracy of these simplified models has not yet been quantified over a range of possible design cases. This paper will quantify how increasing levels of modelling simplifications decrease the accuracy of PCB FE models. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction During the launch phase, spacecraft electronic equipment is subjected to a very harsh vibration environment. If the equipment is not designed to withstand this vibration, the components soldered to the printed circuit board (PCB) may fail. The current process used to ensure that failure does not occur is a trial and error approach mixed with a certain amount of subjective experience, this approach necessitates that a prototype of the board be manufactured and subjected to a representative environment. Generally this approach is cost effective as most prototypes do not fail the vibration ∗ Corresponding author.
E-mail address: [email protected] (R.A. Amy). 0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.01.037
qualification test, but if a prototype does fail the resultant iterative trial and error process can become very costly and time consuming. One solution is to analytically predict the failure by the following two-stage physics-of-failure (PoF) approach. The first stage calculates the vibration response of the board through a finite element (FE) model of the PCB/component system, incorporating some simplifications and assumptions to speed up the modelling process. The second stage relates this calculated response to some pre-determined component failure criteria, to show whether or not the component can withstand the predicted response (e.g. curvature or acceleration). The principal short-coming of the first stage of the PoF process is that the accuracy of the process is unknown, thus the confidence in the failure criteria and consequent reliability estimates cannot be assessed.
R.A. Amy et al. / Acta Astronautica 65 (2009) 192 – 201
This work will illustrate a process to determine this error dependant on the simplification assumptions used. The process is a Monte Carlo simulation of randomly created PCBs at various levels of simplification, with each run being compared to an unsimplified benchmark case. It will then be possible to see the error statistics for each level of simplification. In addition to examining how the relative error varies within each level of simplification, it is also useful to find the correlation of relative error with the geometry and mechanical properties (input variables) of the PCB/component systems, for example, it might be possible to accept a higher level of simplification for smaller or less heavily populated boards. The main practical uses of this process are either to determine an appropriate level of model simplification before a modelling commences or to determine the expected level of error for a pre-existing model. In the first case an allowable error would be given, along with the geometry of the proposed PCB, from this data a simplification regime could be chosen. In the second case a model would be given, along with the levels of simplification used in its creation, from this data the expected error could be calculated. In addition to these two uses it is also possible to gain insight into which geometry and mechanical properties correlate with high stress on components, this knowledge can be easily used to increase the reliability. Essentially the process is a sensitivity analysis of FE models of PCBs, with the inputs being the geometry and mechanical properties. The output is the relative error of the deformed shape with respect to a benchmark case, the relative error is not an absolute value, only the ratio of the simplified system to the benchmark case. Because of these assumptions no experimental analysis is considered necessary, as it would only be measuring the accuracy of the benchmark case, which in this case is not of importance. What is of importance here is the ratio of the results due to the simplification and input variables. 2. Current state of the art Attaching components to a PCB will make the underlying PCB seem to behave as if it has increased density and stiffness properties. If this increase is known for each component, an FE model can be made with the increased density and stiffness properties at each expected component location. To simplify this approach the usual process is to sum the additional mass and stiffness increases provided by the attached components and “smear” this averaged property over the entire FE model
193
of the PCB. This “smearing” process has been previously validated for some specific cases of electronic equipment [1,2], where the validation criteria was error in natural frequencies and the model assurance criterion (MAC). Another area of simplification is in the specification of the model’s edge rotational stiffness; due to lack of information this value is usually modelled as zero (i.e. simply supported). It has been noted that this simplification reduces the accuracy in predicting the overall response much more significantly than property smearing, especially in cases where the edges are firmly clamped [3,4]. Once the response of the PCB to vibration input has been calculated it must be compared to some preexisting failure criteria. Early work focused on the overall RMS acceleration input as the primary cause of failure [5,6]; however, it has been recognised that the individual component local acceleration environment is much more important in contributing to failure [6]. From similar observations, other research has tried to improve the state of the art by considering failure due to the local acceleration environment experienced by each individual component [7]. Recent work has considered the local PCB curvature as the failure criteria as opposed to the local acceleration. This approach is especially pertinent for surface mount technology (SMT) components [6,8–10]. This work improves upon the current state of the art by considering the relative error in curvatures and displacements, as opposed to previous research which only considered the MAC. MAC is deficient as it is only a measure of the overall shape difference of eigenvectors, not error of local curvature. The use of curvature as an error variable shows that although the overall shape may be very similar, small local differences around components may lead to significant error in the stress experienced by components. Furthermore the results of this work are valid over a larger range of possible cases than previous research; this is achieved by using a Monte Carlo approach with many different combinations of geometry and component locations. Finally the effect of error in the rotational edge stiffness is also examined; this focuses on the error induced by modelling the edge rotational stiffness as zero. 3. Random creation of models The Monte Carlo simulation included several hundred individual runs, for each of these runs the following input variables are used to randomly create a PCB FE model: length of first side, ratio of second edge,
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thickness of PCB and edge rotational stiffness. These variables are in various statistical distributions pertinent to the case being studied. The Young’s modulus and density of the unpopulated PCB are assumed to be constant across all models. It is now possible to populate these models with components. This is achieved by first dividing the components into three separate categories: Light components: This is intended to model small components such as resistors or transistors, the length and mass increase of the components is small, the stiffness increase is negligible. Surface mount technology: The SMT category symbolises components such as quad flat pack (QFP), ball grid array (BGA) and pin grid array (PGA), which are generally have a longest edge of about 10.30 mm, and have an increased density and stiffness ratios are in proportion to the length and inversely proportional to the thickness of the PCB to which they are attached. Heavy components: This category is intended to reflect large components such as transformers, large power capacitors and resistors. The density and stiffness ratios are proportional to component length. Each run of the Monte Carlo simulation has different weightings of the different component categories; this allows the effect of different types of components to be assessed. Using the component type weighting the FE model is randomly populated with components, with the size of each component falling within distributions appropriate to the component type. The stiffness and density increases are calculated by statistical distributions that are functions of size, component type and PCB thickness. The stiffness and density increase are achieved by modifying the local PCB Young’s modulus and density to reflect the component contributions. The penultimate step of creating the FE models is to apply the boundary conditions. The displacement degrees of freedom of the edge of the PCB are not considered, as this has been shown to be a safe assumption. The rotational degrees of freedom in the in-plane directions are set to random values between 0% and 100%, where 0% is intended to reflect a simply supported BC and 100% a clamped BC, this is based on the definitions from [4]. The final step in creating the FE models is to create the mass and stiffness matrices, this is achieved using the quad4 element type from the OpenFEM element library.
4. Levels of simplification There are two categories of simplification that are used, property smearing; where the mass or stiffness due to the addition of components is averaged over the entire PCB, and boundary condition (BC) simplification; where the rotational stiffness of the edge of the PCB is assumed or estimated. The levels of property smearing used are: Mass smearing: The area weighted density of the components are summed and averaged over the entire surface of the board. Stiffness smearing: As with mass smearing but stiffness instead. Mass and stiffness smearing: The combination of the previous two levels of smearing. Unpopulated: The simplest level of modelling assumes the PCB/component system can be modelled as the unpopulated PCB, i.e. completely ignore the mass and stiffness effects of the components. Considering the boundary condition simplification, when no other information is available it is usual to model the rotational stiffness as zero. As mentioned previously the rotational edge stiffness is given a fixity value between 0% to 100%. For more information see [3,4].
5. Overall creation and solution process It is appropriate at this point to re-iterate the whole process, which is as follows: 1. Randomly create a set of variables from pre-defined distributions; variables are edge length, edge ratio, thickness, edge stiffness. 2. From these variables create a PCB layout. 3. Randomly locate components on the FE model, where the density, probability and size of the components are within pre-defined distributions. 4. Give each component a density and stiffness ratio increase, where the ratios are dependant on the component type, size and PCB thickness. 5. Solve this initial model to create the “benchmark” case. 6. Apply the various levels of smearing to create the simplified models. 7. Solve each simplified case and compare the results with the “benchmark” case.
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8. Set the edge rotational stiffness to zero and repeat the last two steps. 9. Repeat the whole process many times (700 in the present case study). The models are created and solved in the MATLAB environment using the OpenFEM element library and solvers. The out of plane deformation is found for a unit acceleration input. The PCB curvature is subsequently calculated from the deformation at every node on the FE model, in this case study the nodes are spaced at 5 mm intervals. Comparing the percentage error at each node to its relative node on the benchmark case gives the relative curvature error, , across the model. Once this is calculated in both the x and y directions it is possible to find the statistical measures of , such as maximum, minimum, mean, interquartile range, etc. The same process is repeated for the out of plane deformation, this shall be referred to as z. The eigenvalues and eigenvectors are also calculated to examine the MAC and relative frequency error, this step is mainly for completeness and to allow further insight into the results.
6. Discussion of the results The measure of that will be used in this case study will be the 75th percentile (75th ); this is a measure of the range of the data. The data may not be centred
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around zero, to account for this the mean is added to the half the 75th percentile range. This value was used as it is robust to outliers and can also be used to approximate (standard deviation). Fig. 1 is a plot of 75th against the different levels of smearing, the first observation that can be made of this graph is that the simplification levels that model the BCs as simply supported have a much larger spread of results. This point is clarified in Fig. 2, which includes a third dimension showing the amount of edge rotational stiffness simplification that has occurred. This three dimensional plot shows that the accuracy of the results are very sensitive to errors in specifying the edge rotational stiffness, especially in those cases where the original stiffness was very high (i.e. those cases that had more to lose from simplification), causing the model to suffer a large loss in percentage edge fixity. The sensitivity of the models to the edge rotational stiffness is considerable and tends to dilute the effect of other variables, thus for the rest of this analysis it will be assumed that the edge rotational stiffness can be found and modelled precisely. This will allow the sensitivity of the modelling to other variables to be studied more easily, although it should be kept in mind that large amounts of simplification in edge stiffness will results in large errors in the model. Fig. 3 plots the first four levels of smearing (i.e. those that have no simplification of the edge rotational stiffness) against the ratio of the unpopulated PCB stiffness to the actual stiffness ratio. From this graph it can be
Fig. 1. Smearing level plotted against 75th , “both” refers to joint density and stiffness smearing, note the large variation in the last four levels due to the simplification of the boundary conditions.
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Fig. 2. Smearing level plotted against 75th and edge rotational stiffness, only the four levels of smearing that simplify BCs were plotted.
Fig. 3. Smearing level plotted against curvature delta and ratio of populated to unpopulated stiffness.
seen that there is a distinct positive correlation between the accuracy and the stiffness ratio, although the correlation is stronger for some levels of simplification than others. Fig. 4 shows this for the specific case of stiffness smearing. The correlation of is stronger with stiffness smearing than density smearing; this suggests that smearing the stiffness has a greater negative impact on curvature accuracy than smearing the density. It should be apparent from the large number of input variables, that this trial and error approach to finding the correlation of the input and output variables is a time
consuming process. To reduce the amount of time spent doing this, it is possible to create a matrix that lists the amount of correlation of all the variables against one another. In this matrix the first third of the variables were input variables (see Fig. 5), the last third were various measurements of error and the middle group represent measurements of MAC and fundamental frequency errors. If this matrix is plotted graphically with the correlation value in each cell represented as a colour, then the variables that correlate with each other immediately become obvious. Figs. 5 and 6 illustrate these
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Fig. 4. Stiffness ratio plotted against curvature delta for the stiffness smearing case.
Fig. 5. Cross-correlation of variables for the stiffness smeared case, each number is an index that relates to a specific variable, with x and y axes plotting the same variables. Note that the diagonal of the matrix shows perfect correlation, this is due to the variables being correlated against themselves. See Fig. 6 for further clarification.
correlation plots. In this case it can be quickly seen that the variables that strongly correlate with error are the PCB thickness and the stiffness and density ratios, to a lesser extent other variables also correlate with error. It is now possible to concentrate solely on the effect of
these input variables on the , the following discussion will focus on these variables. However, if no significant correlation are found between any of the input and output variables, only the global statistics (such as those in Fig. 1) can be analysed.
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Fig. 6. Magnification of lower right corner of Fig. 5, several variables have been referenced, the shade of each square represents the correlation between two variables, with darker shades indicating higher correlation.
With these observations it is possible to analyse only those values that strongly correlate with failure. Fig. 7 shows the aforementioned variables plotted together with colour representing the at each discrete point. The resulting points create a three dimensional surface, from which the expected error can be calculated. In a similar was to the previous example it is possible to plot thickness as the fourth (colour) variable, as in Fig. 8. From this it is possible to select a specific thickness (as would very likely occur during the design of a PCB), and focus on data points that relate to this specific thickness of interest whilst ignoring data points relating to other thicknesses. Fig. 9 shows this selective process, the only data points that are plotted are those for the thickness of interest (1.6 mm in this case). If the stiffness and density ratios are known, it is possible to directly assess the expected accuracy of the results. 7. Observations The following observations were noted during the case study, although these observations may not be valid in a different case that uses different input values the implications are the same. The sensitivity of the results to edge rotational stiffness: In agreement with past research [3,4] the results are very sensitive to error in specifying the edge rotational stiffness.
Some variables seem to have little correlation with error: In the cases studied here the results are insensitive to some input variables, such as ratio of edge lengths, whilst other variables show only very small correlation or the correlation depended on the level of simplification. The correlation of stiffness and density ratios with error: It was noted that the stiffness and mass ratios show a strong positive correlation with , this agrees with observations in previous research [2]. Furthermore it is noted that the out of plane displacement are sensitive to simplification of the mass properties, which is in contrast to the curvature, which is more sensitive to simplification of the stiffness properties. Thicker PCBs are less sensitive to the effect of components: Due to fact that thicker PCBs are less receptive to the additional component mass and stiffness, the accuracy of models of thick PCBs are more accurate than those of thin PCBs. In the cases studied those PCBs over 2 mm thick which only composed of light and SMT components all showed curvature error deltas within 5%.
8. Sources of modelling error The following variables were not considered in the case study illustrated, although they could be included in further studies if they were deemed significant.
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Fig. 7. Stiffness and density ratios plotted against PCB thickness, for stiffness smearing case, points are grouped by errors.
Isotropic PCB properties: Real-life PCBs are usually anisotropic due to the lay-up, in the case study the PCB properties were assumed to be isotropic as the lay-up direction might not be known. Anisotropy could be easily included in the model if deemed necessary. Homogeneous density and stiffness under components: The density and stiffness increase due to each component is assumed to be constant in the area underneath each component, further analysis may find a distributed stiffness, but it is assumed that this is a safe assumption. Homogeneous edge rotational stiffness: The rotational edge stiffness was modelled as continuous and constant for all edges, real-life situations are more likely to exhibit discrete locations of stiffness which vary from edge to edge. The homogeneous assumption was used due the excessive amount of runs that would be required to incorporate all different possible combinations of edge stiffness. It would be feasible to model this if really necessary, especially if other input variables were constant.
Zero component height: For small components and those of SMT type, this is a safe assumption, as these small components have negligible height above the PCB. Equipment with higher percentage of large tall components is more likely to be affected by this assumption, as the rocking motions associated with such components may cause stresses that are not modelled. Low frequency change: It is assumed that the change in frequencies is low, as otherwise it is possible that the modes with a significant frequency change may be actually occur at frequencies with a more intense vibration input spectrum (if the vibration input is assumed to be shaped). For the first three levels of simplification studied here, the maximum variation in frequency was typically within 5%. Whereas, if the board was modelled as unpopulated the variation could be as high as 20%, whilst for large amounts of edge stiffening simplification the variation rose to 30%. Ultimately it should be noted that for shaped vibration input, the statistics of frequency change should be taken into account when choosing a simplification regime.
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Fig. 8. Stiffness and density ratios plotted against error, for stiffness smearing case, points are grouped by thickness.
Fig. 9. Stiffness and density ratios plotted against error, for stiffness smearing case at 1.6 mm.
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9. Conclusion A process has been illustrated that allows the relative error of PCB FE models to be assessed; this is calculated at various levels of model simplification. The resulting data sets can then be analysed using the techniques suggested, eventually providing the expected relative error for given levels of smearing, which can be further decomposed into errors for various input variables. Some important observations that were made during the data analysis were that the curvature was more sensitive to simplification of the stiffness properties, whilst overall response was more sensitive to density smearing. The sensitivity of modelling accuracy to large amounts of simplification in the edge rotational stiffness was also highlighted. In addition to this, the data can also provide valuable insight into the nature of the response of a particular board, and how simple modifications to only a few variables can reduce the response with very little cost. Acknowledgement The author would like to warmly thank Surrey Satellite Technology Limited (SSTL) for their vital contribution to this research. References [1] J.M. Pitarresi, A. Primavera, Comparison of vibration modeling techniques for printed circuit cards, ASME Journal of Electronic Packaging 114 (1991) 378–383.
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[2] J.M. Pitarresi, D.V. Caletka, R. Caldwell, D.E. Smith, The “Smeared” property technique for the FE vibration analysis of printed circuit cards, ASME Journal of Electronic Packaging 113 (1991) 250–257. [3] G.H. Lim, J.H. Ong, J.E.T. Penny, Effect of edge and internal point support of a printed circuit board under vibration, ASME Journal of Electronic Packaging 121 (2) (1999) 122–126. [4] D.B. Barker, Y.S. Chen, Modeling the vibration restraints of wedge-lock card guides, ASME Journal of Electronic Packaging 115 (2) (1993) 189–194. [5] J. Lau, L.M. Powers-Maloney, J.R. Baker, D. Rice, B. Shaw, Solder joint reliability of fine pitch surface mount technology assemblies, IEEE Transactions on Components, Hybrids, and Manufacturing Technology 13 (3) (1990) 534–544. [6] R.S. Li, L. Poglitsch, Fatigue of plastic ball grid array and plastic quad flat packages under automotive vibration, in: SMTA International, Proceedings of the Technical Program, 2001, pp. 324–329. [7] S. Liguore, D. Followell, Vibration fatigue of surface mount technology (SMT) solder joints, in: Annual Reliability and Maintainability Symposium 1995 Proceedings (Cat. no. 95CH35743), 1995, pp. 18–26. [8] D.S. Steinberg, Vibration Analysis for Electronic Equipment, Wiley, 2000. [9] H. Wang, M. Zhao, Q. Guo, Vibration fatigue experiments of SMT solder joint, Microelectronics Reliability 44 (7) (2004) 1143–1156. [10] S. Shetty, V. Lehtinen, A. Dasgupta, V. Halkola, T. Reinikainen, Fatigue of chip scale package interconnects due to cyclic bending, ASME Journal of Electronic Packaging 123 (3) (2001) 302–308.