3.1 INTRODUCTION
This chapter describes the methodology for performing the modal analysis of a printed circuit board used in a hand held electronic components by analytical and experimental methods.
3.2 NEED FOR PERFORMING MODAL ANALYSIS
In the past two decades, modal analysis has become a major tool in the quest for determining, improving and optimizing dynamic characteristics of engineering structures. Not only has it been recognized in mechanical and aeronautical engineering, but also modal analysis has discovered profound applications for civil and building structures, biomechanical problems, space structures, acoustical instruments, electronic components, and nuclear plants.
vibration are inherent to a dynamic system and are determined completely by its physical properties (mass, stiffness, damping) and their spatial distributions. Each mode is described in terms of its modal parameters: natural frequency, the modal damping factor and characteristic displacement pattern, namely mode shape. The mode shape may be real or complex. Each corresponds to a natural frequency. The degree of participation of each natural mode in the overall vibration is determined both by properties of the excitation source(s) and by the mode shapes of the system.
Modal analysis embraces both theoretical and experimental techniques. The theoretical modal analysis anchors on a physical model of a dynamic system comprising its mass, stiffness and damping properties. These properties may be given in forms of partial differential equations. A more realistic physical model will usually comprise the mass, stiffness and damping properties in terms of their spatial distributions, namely the mass, stiffness and damping matrices. These matrices are incorporated into a set of normal differential equations of motion. The superposition principle of a linear dynamic system is used to transform these equations into a typical eigen value problem. Its solution provides the modal data of the system.
Modern finite element analysis empowers the discretization of almost any linear dynamic structure and hence has greatly enhanced the capacity and scope of theoretical modal analysis. On the other hand, the rapid development over the last two decades of data acquisition and processing capabilities has given rise to major advances in the experimental realm of the analysis, which has become known as modal testing.
PCB are the base for understanding the dynamic response characteristics of the PCB and the solder joints. For a particular IC package, PCB and its mounting conditions, the natural frequencies and the mode shapes are its inherent characteristics which can be obtained by modal analysis. These dynamic characteristics in turn decide the response of the electronic equipments when subjected to vibration loads or when they are subjected to drop impact conditions. The response of the electronic equipment in turn determines the vulnerable or critical areas in the equipment that are subjected to high stresses and may fail prematurely during its service life. Designers can then concentrate on these areas to reduce the stresses developed in the design stage itself which ensures that the equipment performs its intended function for the specified time rather than failing prematurely.
from the accelerometer. The methodology used in the modal analysis of the PCB is shown in Figure 3.1.
Figure 3.1 Methodology adopted in the modal analysis of the PCB Modal analysis of a PCB
Finite Element Method
Develop a FE model of the PCB
Apply material properties and boundary conditions
Solve the FE model for modal analysis
Extract the natural frequencies and mode shapes
Experimental method
Fabricate a PCB with packages mounted
Mount the PCB on a rigid fixture
Apply an impulse using an impact hammer
Acquire the PCB response using an accelerometer
Calculate the FRF from the signal acquired from the
impact hammer and accelerometer
Extract the mode shapes, natural frequencies and damping ratios from the FRF
- Location of the packages mounted on PCB
location where the packages are mounted and the locations where the PCB is supported is shown in Figure 3.2 and the actual PCB custom made for this research is shown in Figure 3.3. The BGA package mounted on the PCB has a size of 2.5 x 2.5 x 0.74 mm and is mounted on to the board by means of 15 solder balls. The details of the BGA package are shown in Figure 3.4.
Figure 3.3 Printed Circuit Board with BGA packages
Figure 3.4 Detailed layout of BGA Package
Figure 3.5 shows the FE model of the PCB together with the packages. The boundary conditions applied to the FE model are the same as that of the actual PCB.
Figure 3.5 Finite Element (FE) Model of the PCB with BGA Package
The material properties of the various components in the PCB are listed in Table 3.1. One of the important assumptions in the modal analysis by FE method is that the materials are assumed to be linear isotropic in nature even though the PCB itself is a composite material made of FR4/Epoxy. This assumption is important because the system as a whole is considered as a linear system which is the precondition for performing modal analysis.
BGA Package
Table 3.1 Material Properties of PCB and packages used in FEA (Pecht et al 1999) S.No Component Young’s modulus, E (Pa) Poisson’s ratio, Density, (kg/m3) 1. PCB 1.7 x 1010 0.35 2200 2. Package 2.2 x 1010 0.29 2100 4. Solder ball 3.2 x 1010 0.38 8440
The maximum frequency of vibration to which hand held electronic devices are exposed is found to be 1000 Hz (JESD22-B103B, 2006). Hence, it is decided to extract the natural frequencies with in 1000 Hz from FEM. The first, second, third and fourth mode shapes and the corresponding natural frequencies obtained from the FE model are shown in Figures 3.6, 3.7, 3.8 and 3.9 respectively and the same is tabulated in Table 3.2.
Figure 3.7 Second natural frequency (573.483 Hz) and mode shape of the PCB
Figure 3.9 Fourth natural frequency (876.465 Hz) and mode shape of the PCB
Table 3.2 Natural frequencies from FE Model of the PCB
Mode Natural frequency (Hz)
1 341.08
2 573.48
3 821.48
4 876.47
3.4 MODAL ANALYSIS BY EXPERIMENT METHOD
Experimental modal analysis in the frequency domain can be classified into three different test methods based on the number of FRFs which are to be included in the analysis (Ewins 2001). The simplest of the three methods is referred to as SISO (Single Input, Single Output) which involves measuring a single FRF for a single input given. A SISO data set is made of a set of FRFs which are measured individually but sequentially. The second test method is referred to as SIMO (Single Input, Multiple Output). This refers to a set of FRFs measured simultaneously at different locations for a single input given at a specific location. The third method is referred to as MIMO (Multiple Input, Multiple Output) in which the FRFs at various points are measured simultaneously while the structure is excited at several points simultaneously. Since the PCB is a light weight structure, mounting several accelerometers to acquire FRFs simultaneously will result in erroneous natural frequencies as the mass of the accelerometers will affect the natural frequencies of the PCB. Hence in this research, SISO method was adopted to perform experimental modal analysis.
3.4.1 Experimental setup
The experimental setup for performing the modal analysis consists of the following equipments:
(b) Accelerometer (c) Signal conditioner (d) Data Acquisition System (e) Base plate fixture
The schematic representation of the experimental setup is shown in Figure 3.10 and the actual test setup is shown in Figure 3.11.
The impulse/impact hammer (PCB Piezotronics Model 086C03) is used in the modal analysis for giving the necessary excitation to the PCB. The impact hammer is provided with several types of heads typically made up of plastic, steel or rubber to give different excitation levels depending upon the structure being tested. The accelerometer is used to pick up the vibration signals from various locations on the PCB. To minimize the effect of the mass of the accelerometer on the natural frequencies of the PCB a miniature accelerometer (B&K Model 4517) weighing 0.6 gm was used in the test. The PCB is clamped to an aluminium base plate fixture at its four support locations and the base plate is in turn clamped rigidly to a table to isolate the PCB from external excitations.
Figure 3.10 Schematic of the experimental set up for modal analysis
Figure 3.11 Experimental set up for modal analysis
It acts as a bridge between the sensors and the computer. It is capable of acquiring eight simultaneously sampled analog inputs at a rate of 102.4 kS/s. NI PXI-1042Q is a stand alone, portable computer operating on Windows XP. The chassis supports the LabVIEW 8.6 software which is used for analyzing the input signals from the accelerometer and the impact hammer and to calculate the FRFs based on the input signals. It is also capable of displaying acquired signals and the calculated FRFs either in time domain or in the frequency domain.
3.4.2 Experimental procedure
The modal testing based on SISO method can be performed in two ways. One is called the roving hammer technique and the other is roving accelerometer technique (Bruel & Kjaer, 2003). Both the techniques rely on the principle of Maxwell’s reciprocity theorem (Rao S.S., 2004). According to this principle, for a linear system, the FRF measured at a point ‘j’ for an excitation given at a point ‘i’ (Hij) will be equal to the FRF measured at point
‘i’ for an excitation given at point ‘j’ (Hji).
i.e. Hij = Hji
in the nodal line of the PCB mode shapes. To determine the locations on the PCB for giving excitation, the PCB was divided into small segments. The layout of the PCB after it was divided into smaller segments; the locations for exciting the PCB using impact hammer and the location for mounting the accelerometer were shown in Figure 3.12. The location of the accelerometer was decided by analyzing the mode shapes of the PCB obtained from FEM. Altogether, the PCB was excited at 48 different points sequentially in the order given in Figure 3.12 and the response was measured by the accelerometer for each excitation.
Figure 3.12 Layout of the PCB showing the excitation and response points 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 39 30 31 32 33 34 35 36 37 38 40 47 41 42 43 44 45 46 48
- Location of the packages mounted on PCB - Location of excitation by impact hammer
The next step in the modal testing is to determine the head to be used in the impact hammer for exciting the PCB. The impact hammer was provided with various types of heads made of different materials and was detachable for using in different types of structures. The head to be used for exciting the PCB should excite as many natural frequencies as possible. This depends on the time duration of the exciting impulse. If the time duration of the exciting impulse is too large, then the frequency spectrum curve which shows the variation of the exciting impulse with respect to the frequency will decrease rapidly thereby making the measured FRF unreliable.
The decay in the spectrum curve for the frequency range of interest for the plastic head was around 15 dB. For the other heads the decay is around 20 dB. Hence, for performing the modal test on the PCB, the plastic head was used in this research.
3.4.3 Experimental results
The modal test on the PCB was carried out by hitting the PCB at the specified points sequentially from point 1 to point 48 as shown in Fig 3.9 and the response is measured for each of the given impulse. At each point three impacts were given and the results were averaged to reduce the effect of noise in the acquired signal. In experiment modal testing, one of the tools used to ensure the quality of the acquired signal is coherence (Ewins 2001). Coherence indicates how much the measured response is correlated to the input excitation. If there is another signal present in the response, either from noise or from other signal, the quality of the response measured will be poor. Figure 3.14 shows the measured excitation and response.
Figure 3.14 Schematic representation of measurement of input excitation (hammer) and the response (accelerometer)
Applied excitation Structure under modal
test
Frequency response function (H(f)) can then be defined as:
H(f) = SAB(f) / SAA(f) (3.1)
where SAB(f) = cross power spectrum of A and B
SAA(f) = power spectrum of A
Coherence function(2) can then be defined as:
2 = ( SAB(f)) 2
/ SAA(f)* SBB(f) (3.2)
where SBB(f) = power spectrum of B
Coherence can have a maximum value of 1 and a minimum value of 0. Coherence value of 1 indicates that the response measured is entirely due to the given input excitation and value of 0 indicates that the measured response is entirely due to some other excitation than the given excitation. Coherence function was measured several times during the modal impact test and some of the measured coherence is shown in Figure 3.15.
[
Figure 3.16 FRF measured at one particular location on the PCB showing the real and imaginary curves
According to the theory of modal analysis (Ewins 2001), the natural frequencies will occur at the peak of the imaginary curve and at the middle of adjacent positive and negative peaks of the real curve. From Figure 3.16 the first three natural frequencies measure by the FRF were found to be 340 Hz,
615 Hz and 840 Hz respectively. The fourth natural frequency was not excited by the given impulse and hence was not shown in the FRF (Figure 3.16). Similar trend was reported in various literature were experimental modal testing was not able to excite all natural frequencies within a specified range (Li 1999), (Chen 2006).
3.4.4 Extraction of modal parameters
Once the modal test was completed, the next step is to extract the modal parameters namely the natural frequencies, the mode shapes and the damping ratios. These parameters are to be extracted by post processing the FRFs captured in the modal test. For post processing, DIAMOND software (Diamond 1997) was used. DIAMOND is a graphical user interface toolkit developed using Matlab by Los Alamos National Laboratory, Los Alamos, USA. DIAMOND stands for (Damage Identification And MOdal aNalyis for Dummies) and it has the capability of analyzing modal data from single or multiple reference experiments to determine natural frequencies, damping ratios and mode shapes by using a variety of modal curve fitting algorithms.
Figure 3.17 Geometry of the PCB created in DIAMOND software
Figure 3.19 Second natural frequency (615 Hz) and the extracted mode shape
3.5 COMPARISON OF RESULTS
The extracted mode shapes were compared with mode shapes obtained from FEM as shown in Table 3.4 and were found to be in good agreement. Similarly, the extracted natural frequencies from experiments were then compared with that obtained from FE method and is shown in Table 3.5. The next step in the experimental modal analysis is the extraction of damping ratios. For extracting the damping ratios, half power band width technique was used.
According to this technique (Rao 2004), the damping ratio is given by the expression
1 2 r 2 (3.3)
where 1, 2 are frequencies at which the magnitude of the frequency response curve is (1/2) times the peak response obtained at the natural frequency r. The half power bandwidth for the first natural frequency is
Table 3.4 Comparison of mode shapes obtained from FEM and experiments
Mode
Number Mode shape from FEA
Mode shape from experiment
1
2
3
3.6 CONCLUDING REMARKS
In this chapter, a comprehensive method for performing modal analysis on a PCB by both finite element and experimental methods was described. The natural frequencies and mode shapes were extracted from the FE model of the PCB with packages mounted on it. The first four natural frequencies obtained from the FE model were found to be 341.08 Hz, 573.48 Hz, 821.48 Hz and 876.47 Hz respectively.
Experimental model testing was conducted on a custom made PCB using SISO technique. The PCB was excited by an impulse hammer and the response of the PCB was measured using an accelerometer mounted on the PCB. The measured FRFs were then used to extract the natural frequencies, mode shapes and damping ratios of the PCB. The extraction of these dynamic characteristics of the PCB was done using DIAMOND software. The first, second and the third natural frequencies obtained from the experimental method were found to be 341 Hz, 615 Hz and 830 Hz respectively. The corresponding damping ratios were found to be 0.012, 0.0072 and 0.0032 respectively.
experimental method. The maximum percentage of variation between the two methods was found to be 6.7% occurring at the second natural frequency.
Figure 3.21 Half power bandwidth for calculating the damping ratio at the first natural frequency
