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Optimal laser-cutting parameters for QFN packages by utilizing artificial neural networks and genetic algorithm
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 270–283
journal homepage: www.elsevier.com/locate/jmatprotec
Optimal laser-cutting parameters for QFN packages by utilizing artificial neural networks and genetic algorithm Ming-Jong Tsai a,∗ , Chen-Hao Li a , Cheng-Che Chen a,b a
Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, No. 43, Keelung Road, Section 4, Taipei 10672, Taiwan b Gallant Precision Machining Company, Taipei, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, a multiple regression analysis (MRA) and an artificial neural network (ANN)
Received 30 July 2007
were employed to build a predicting model for cutting Quad Flat Non-lead (QFN) packages
Received in revised form
by using a Diode Pumped Solid State Laser (DPSSL) System. The predicting model includes
31 October 2007
three input variables of the current, the frequency and the cutting speed, and six cutting
Accepted 27 December 2007
qualities of depths of the cutting line, widths of heat affected zone (HAZ) and cutting line for epoxy and for copper-compounded epoxy. After the training process from 27 sets of training data including input data and its output qualities, the average training error is 0.822% by
Keywords:
using a back-propagation (BP) neural network with Levenberg–Marquardt (LM) algorithm,
QFN package
which leads to the best results. The testing accuracy is then verified with extra 14 sets
Laser cutting
of experimental data and the average predicting error is 1.512%. The results show that the
Multiple regression analysis (MRA)
ANN model has the predicting ability to estimate the laser-cutting qualities of QFN packages.
Artificial neural network (ANN)
Finally, a genetic algorithm (GA) is applied to find the optimal cutting parameters that lead
Levenberg–Marquardt (LM)
to least HAZ width and fast cutting speed with complete cutting. The optimal combination
algorithm
found is the current of 29 A, the frequency of 2.7 kHz and the cutting speed of 3.49 mm/s. The
Genetic algorithm (GA)
GA is helpful to determine the ideal laser-cutting parameters in order to meet the desired cutting qualities and to avoid unnecessary adjustments in the subsequent cutting process. © 2008 Elsevier B.V. All rights reserved.
1.
Introduction
A Quad Flat Non-lead (QFN) package is one of the main semiconductor packaging technologies. It consists of a plastic encapsulated package with a copper lead frame substrate. This packaging technology, which has many advantages, such as less size and weight, good electrical performance and high speed and frequency, has been applied widely in many products (Chen et al., 2003). In cutting QFN ICs, the conventional technology using diamond saw has the advantage of high-
∗
Corresponding author. Tel.: +886 2 27376286; fax: +886 2 27301265. E-mail address: [email protected] (M.-J. Tsai). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.12.138
speed cutting but tends to make the QFN ICs breakable and easy to crack. This is an area where laser cutting may offer a solution by providing a technique, which enables IC cutting to be carried out with less width of cutting line and less heat affected zone (HAZ) (Tsai et al., 2005; Li et al., 2007). Nowadays lasers are widely used in manufacturing and automotive industries, such as laser cutting, welding, marking, and drilling, in order to manufacture low cost, high quality products in short time. Many laser control parameters affecting laser-cutting qualities have been investigated,
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including the effect of laser-cutting speed on surface temperature (Rajendran and Pate, 1988), on surface quality (Neimeyer et al., 1993), and on surface roughness (Rajaram et al., 2003), and the effect of laser frequencies on HAZ (Kaebernick et al., 1998). In the meanwhile, many efforts have been made in the literature to determine the optimal values for these laser control parameters (Olsen, 1983; Steen and Kamalu, 1983; Hamoudi, 1996; Nagarajan, 2000). Nevertheless, the mechanism behind laser cutting is very complicated and process dependent, making it difficult to predict laser-cutting quality via analytical formulae whose validity often applies to limited range of processes and cutting conditions. Consequently, there is a need for a tool that will allow the evaluation of laser-cutting quality before the actual cutting process. Multiple regression analysis (MRA) is one method of statistics, which is utilized to build a mathematical model that describes the relation between input factors and output factors. This method is widely applied in many engineering, including the model of surface finish in electro-discharge machining (Petropoulos et al., 2004) and elastic properties of intact rocks (Karakus et al., 2005). Artificial neural networks (ANN) is an empirical modeling tool analogous to the behavior of biological neural structures and can identify highly complex relationships by using input–output data (Benardos and Vosniakos, 2002; Cheng and Lin, 2000; Dobrzanski et al., 2005; Singh et al., 2007; Bariani et al., 2004; Chelani et al., 2002; He et al., 2005). A common ANN model is established in a training process by using backpropagation (BP) with the gradient descent (GD) algorithm. To conquer the problem of slow convergence encountered by BP with GD, many improvements have been proposed, such as scaled conjugate gradient (SCG) algorithm, quasi-Newton (QN) algorithm and Levenberg–Marquardt (LM) algorithm. The algorithms improve common GD and produce less predicting error. In this paper above four algorithms (GD, SCG, QN and LM) are used to train the predicting model of laser-cutting qualities, respectively, and compare the predicting errors to obtain least errors and accurate predicting values. Genetic algorithm (GA) has been widely applied in many areas, such as control, prediction, forecasting, optimization, differentiation, vision, etc. The use of GA has become popular in recent years to optimize the process parameters and obtain optimal solutions in the engineering, including the optimization of the designing of chemical composition of high-speed steels (HSS) (Sitek and Dobrzanski, 2005), and the die shape in extrusion (Chung and Hwang, 1997), and optimal output
feedback controller (Badran and Al-Duwaish, 1999), and heat sources on a vertical wall with natural convection (Dias and Milanez, 2006), and dual-homing cell assignment problem of the two-level wireless ATM network (Din and Tseng, 2002) and controller parameters of heating ventilating and air conditioning (HVAC) (Huang and Lam, 1997). In this paper, the multiple regression analysis (MRA) and the artificial neural network are utilized to build a predicting model for the six laser-cutting qualities of the QFN packages, respectively. A total 41 sets of experimental data are used to train and verify the ANN model for predicting the cutting qualities. Finally, the genetic algorithm is utilized to find the optimal cutting parameters that lead to less HAZ width and fast cutting speed with the complete cutting.
2. Descriptions of QFN packages and laser-cutting system The internal structure, the cutting path and the microscopic dimensions of QFN packages are described in Fig. 1 (Li et al., 2007). A QFN package is a plastic encapsulated lead-frame based Chip Scale Package (CSP) with the lead pad on the bottom of the package to provide electrical interconnection with the printed circuit board (PCB) (Chen et al., 2003). Several QFN chips are packaged in an array form and must be separated into individual QFN chips before mounting on different printed-circuit boards. Because the whole size of a QFN package is small, the width of the tolerable cutting path between packaged ICs is also very restricted. For a 5 × 5 QFN package the maximum tolerant dimension of the cutting path is 0.26 mm and the thickness is 0.9 mm as shown in Fig. 1 (Li et al., 2007). The patch is made up of two materials, one is the epoxy with a thickness of 0.9 mm and the other is the copper circuit-pad bonding epoxy for which the dimensions of copper and epoxy are 0.24 mm and 0.66 mm, respectively. Fig. 2 sketches the six main parameters (e) governing the cutting quality, including the width (WHAZ ) of (e)
(e)
HAZ, the width (WLine ) and the depth (DLine ) of the cutting line (c) (c) for epoxy, and the three counterparts WHAZ , WLine
(c)
and DLine for
copper circuit-pad bonding epoxy (Li et al., 2007). (e)
(c)
The equipment for measuring the widths (WHAZ , WHAZ ) of (e)
(c)
HAZ and the widths (WLine , WLine ) of cutting line is an electron microscope with measurement accuracy of 1 m. The equip-
Fig. 1 – The cutting path and dimensions of a 5 × 5 QFN package.
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Fig. 2 – Geometrical illustration of six laser-cutting qualities of a QFN package.
(e)
(c)
ment for measuring the depths (DLine , DLine ) of the cutting line is an optical electron microscope with measurement accuracy of 1 m. Due to the irregularity of the widths of HAZ and cutting line, the values of widths along the cutting trajectory are indeed not constant. In order to obtain an accurate average measurement, a mean value of ten-point measurements is used. The depth measurement for the cutting line is taken at four fixed positions and a mean depth can be obtained. In this paper, the laser-cutting system shown in Fig. 3 consists of a diode pumped solid-state laser system (DPSSL; model: Rofin 100D, 1.064 m), a laser-cutting head, an X–Y moving table driven by two servomotors. The moving speed of the X–Y table is defined as the cutting speed in this paper. The current, the frequency and the cutting speed are three laser-cutting parameters adjusted by the controlling panel of the laser system.
3. Main laser control parameters and six cutting qualities In this section, the influences of laser power and cutting speed on the cutting qualities, including the widths and depths of cutting line and the widths of HAZ, will be discussed by using the aforementioned DPSSL system for cutting a QFN package.
3.1.
According to the previous reports (Tsai et al., 2005, 2006; Li et al., 2007), it was known that the dominant laser parameters affecting the cutting quality for a QFN package are the current A, the laser pulse frequency F and the cutting speed V, which are then taken as the main laser control parameters in this paper.
3.1.1.
The current
Using lower current provides lower peak-power for single pulse laser and reduces cutting heat so as to decrease the widths of cutting line and HAZ. However, applying lower current is difficult to cut off the materials effectively. On the contrary, a higher current can generate higher peak-power and release more cutting heat and thus increase the cutting line and HAZ widths. Nevertheless, the cutting process may be more efficient by adopting higher current.
3.1.2.
The frequency
The lower frequency usually leads to higher peak-power of single pulse and gives higher cutting ability. However, the lower frequency also accompanies low-level pulse overlapping and non-continuous power density within a unit length. Therefore, the lower frequency laser cutting tends to produce a noncontinuous cutting line and to generate burn spots along the cutting path. On the contrary, higher frequency cutting has relatively lower peak-power for single pulse and leads to less cutting ability, but it provides higher level pulse overlapping to yield continuous power density within a unit length.
3.1.3.
Fig. 3 – The laser-cutting system for QFN packages.
Main laser control parameters
The cutting speed
Applying lower speed cutting process often accompanies higher level overlapping and continuous power density per unit length and hence tends to cut completely; but it produces more heat and wider HAZ and cutting line. On the contrary, a higher speed cutting process results in narrow HAZ and cutting line at the risk of producing incomplete cutting. In fact, the average output power (Watt) of a laser system is a major factor for providing the required cutting energy. The available output power Y that is measured by using a DPSSL
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system is produced at various combinations of the current A and the laser pulse frequency F. The single laser pulse power K (Joule) is expressed by the following equation (Thawari et al., 2005):
y3j = k30j + Ak31j + Fk32j + V k33j
F , V
Yp = K × S =
Y , V
(2)
where V is the cutting speed and S is also known as pulse overlapping, which is an indication of the smoothness and the continuity of laser cutting. The variations of the power per unit length Yp at various combinations of current A, frequency F and cutting speed V are discussed and illustrated in our previous paper (Li et al., 2007). In order to build the mathematical relations between the three laser control parameter and six cutting qualities, a column vector p containing three input cutting parameters is defined as follow: T
T
p = [A F V] = [current, frequency, cutting speed]
3.2.
y4j = k40j × Ak41j × Fk42j × V k43j
(e)
(c)
t = D(e) Line
(c)
DLine
(e)
WHAZ
(c)
WHAZ
(e)
WLine
(c)
WLine
T
y5j = k50j + eA×k51j + eF×k52j + eV×k53j
y6j = k60j × eA×k61j × eF×k62j × eV×k63j
(4.6)
where yij is jth output cutting quality of ith model (i = 1–6
type
(e)
and
(c)
(e)
(c)
(e)
(c)
j = 1 − 6 = DLine , DLine , WHAZ , WHAZ ,
WLine , WLine ), A, F, V are laser input cutting parameters and kidj is dth regression coefficient at jth output cutting quality of ith model (d = 0–3). And all regression coefficients are estimated for MRA model. In this paper, the analyses were implemented by using Statistics Package for Social Science (SPSS) software package.
4.2.
The ANN model
The structure of the ANN adopted in this paper includes three layers: an input layer, a hidden layer and an output layer, as illustrated in Fig. 4. The input layer receives three external input parameters (A, F, V) from the laser-cutting system and the output layer provides the outputs of the six laser-cutting qualities as follow:
(3.2) (2)
(2)
(2)
(2)
(2)
(2)
(e)
(c)
(e)
(c)
(f1 , f2 , f3 , f4 , f5 , f6 ) = (DLine , DLine , WHAZ , WHAZ , (e)
(c)
WLine , WLine ).
In this section, the relations between laser control parameters for cutting a QFN package and six output cutting qualities were modeled by using MRA and ANN methods.
The MRA models
For MRA modeling, six different multiple regression models that include linear, power and exponential relations are adopted for predictions of six output cutting qualities of a QFN package, respectively. The six MRA models are described as follows: (1) Model 1: First type linear relation (4.1)
(5)
To accommodate the required input and output numbers, three and six neurons must be used in the input and output layers, respectively, i.e., s(0) = 3 and s(2) = 6. The neuron number s(1) in the hidden layer is free to choose, but a best s(1) will be determined later in the next section. The transfer function used in the hidden layer is in the form of log-sigmoid function as follow:
⎡
⎤ ⎡ 1 (1) ⎢ 1 + exp(−n ⎥ 1,q ) (1) (1) f2 (n2,q ) ⎥ ⎢ ⎢ .. ⎥ ⎢ ⎥=⎢ . .. ⎥ . ⎦ ⎢ 1 ⎣ (1)
⎤
(1)
f1 (n1,q )
⎢ ⎢ (1) (1) ⎢ f (nq )=⎢ ⎢ ⎣ f
y1j = k10j + k11j A + k12j F + k13j V
(4.5)
(6) Model 6: Second type exponential relation
4. The MRA and ANN models for laser-cutting qualities
4.1.
(4.4)
(5) Model 5: First type exponential relation
Six output cutting qualities
of cutting line (WLine , WLine ) were described in Tsai et al. (2006), Cenna and Mathew (2005) and some experimental results were reported in Li et al. (2007). A column vector t containing six output cutting qualities of a QFN package is defined as follow:
(4.3)
(4) Model 4: Second type power relation
(3.1)
The theoretical models describing their relationships between the average output power Y, the cutting speed V and the six output qualities including the depth of cutting line (e) (c) (e) (c) (DLine , DLine ), the widths of HAZ (WHAZ , WHAZ ) and the widths
(4.2)
(1)
The other two measures of power dominating the cutting quality include the number of laser pulses per unit length S and the power per unit length Yp (Joule/mm), which can be obtained by following equation: S=
y2j = k21j A + k22j F + k23j V (3) Model 3: First type power relation
Y . F
K=
(2) Model 2: Second type linear relation
(1)
s(1)
(1)
(n
s(1) ,q
)
(1)
1 + exp(−n
s(1) ,q
⎥ ⎥ ⎥ ⎥ , q=1, 2, . . . , 27, ⎥ ⎥ ⎦ ) (6)
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Fig. 4 – The ANN structure for predicting six laser-cutting qualities of a QFN package.
(1)
(0)
where nq = W(1) aq + b(1) is the net input of the hidden layer at qth data indicating input parameters relating to its output (2) qualities (Hagan et al., 1996). As to the transfer functions fj for the output neurons, the linear relation is employed for the last four outputs: (2) (2) fj (nj,q )
=
(2) nj,q ,
j = 3, 4, 5, 6.
(7) (2)
(e)
(2)
(c)
The first two cutting qualities f1 = DLine and f2 = DLine need separate consideration. Because the thickness of QFN package in the present case is 0.9 mm, a cutting depth greater than 0.9 mm means a complete cutting. Accordingly, the outputs of the first and second neurons regarding cutting depth are defined as follow:
(2)
(2)
fj (nj,q ) =
⎧ 0, ⎪ ⎪ ⎨
(2)
n ,
(2)
nj,q ≤ 0 0
(2)
< 0.9 ,
j,q j,q ⎪ ⎪ ⎩ 0.9, n(2) ≥ 0.9 j,q
j = 1, 2.
(8)
With the specified transfer functions f(1) and f(2) , the out(2) puts aq of the ANN regarding the cutting quality predictions can be determined iteratively from the reference (Hagan et al., 1996) along with the iterations of the weights W(1) and W(2) , and biases b(1) and b(2) . The ANN is trained by 27 sets of inputs listed in Table 1 and by the associated measurement outputs recorded in Li et al. (2007). By learning the patterns of these inputs and outputs, the weights and biases are adjusted iteratively during the training procedures (Hagan et al., 1996). At the end of training process, the ANN predicting errors for each
Table 1 – Twenty-seven sets of laser-cutting parameters for the training process Data set 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
Current (A)
Frequency (kHz)
29 29 29 29 29 29 29 29 29 33 33 33 33 33 33 33 33 33 37 37 37 37 37 37 37 37 37
0.5 0.5 0.5 2 2 2 3.5 3.5 3.5 0.5 0.5 0.5 2 2 2 3.5 3.5 3.5 0.5 0.5 0.5 2 2 2 3.5 3.5 3.5
Cutting speed (mm/s) 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5
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Table 2 – The predicting errors of six MRA models using 27 sets of training data Model type
Laser-cutting quality
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
DLine
(e)
DLine
(c)
9.603 6.921* 10.089 11.228 10.836 11.524
18.316 7.602* 19.068 23.867 21.477 25.27
(e)
(c)
WLine
WLine
14.403 13.968 13.564 13.159 21.355 7.482*
3.9225 6.528 4.457 6.448 8.875 3.132*
13.949 13.734 12.984 16.449 26.537 7.713*
WHAZ
(e)
WHAZ
17.021 16.359 16.388 15.532 25.589 7.416*
(c)
(*) Represent the least predicting error; unit: %.
cutting quality are defined as follows: |tj,q − yj,q |
ej,q (%) =
tj,q
× 100%,
j = 1–6
1 ej,q 27
regression equations of six cutting qualities can be estimated as follows: (9.1)
27
ej (%) =
(9.2)
q=1
where ej,q is jth output cutting quality at qth data and is an element of the error vector v defined in the reference (Hagan et al., 1996). Therefore, the average predicting error of the six cutting qualities can be obtained by
eave (%) =
5.
6 1
6
ej .
(10)
j=1
Results and discussions
According to the previous experiments (Li et al., 2007), the lower and upper bounds for every cutting parameter are determined, i.e., current between 29 A and 37 A, frequency between 0.5 kHz and 3.5 kHz, and cutting speed between 0.5 mm/s and 3.5 mm/s. With the established ranges of cutting parameters, the maximum, the minimum and the mean are chosen as the three levels for each cutting parameter. That is to say, the three levels for current are 29 A, 33 A and 37 A; frequency are 0.5 kHz, 2 kHz and 3.5 kHz, and cutting speed are 0.5 mm/s, 2 mm/s and 3.5 mm/s. Table 1 summarizes the laser-cutting parameters and their levels. From the table, 27 sets of various combinations of input cutting parameters are chosen to experiment by using a DPSSL then every combination obtains six cutting qualities. By the way, the experiment of each combination is repeated four times to give an average value. The 27 sets combinations that present three input cutting parameters and six cutting qualities are adopted as training data.
5.1.
The predicting results of MRA models
5.1.1.
Training process of MRA models
A total of 27 sets of training data are employed to train MRA model. Table 2 shows that the predicting errors e1 , e2 , . . ., e6 of the six cutting qualities computed from Eq. (9) at MRA model of six types. From the table, it is observed (e) (c) that the predicting errors of DLine , DLine using model 2 and (e)
(c)
(e)
(c)
WHAZ , WHAZ , WLine , WLine using model 6 are least. The multiple
⎧ (e) DLine = (0.0217 × A) + (0.0915 × F) + ((−0.0529) × V) ⎪ ⎪ ⎪ ⎪ D(c) = (0.0188 × A) + (0.0141 × F) + ((−0.0859) × V) ⎪ ⎪ ⎪ ⎪ Line (e) ⎨ A×0.0414 F×0.2148 V×(−0.278) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
WHAZ = 0.0856 × e
×e
×e
(c) WHAZ (e) WLine (c) WLine
× eF×0.1775
× eV×(−0.2302)
=
0.1098 × eA×0.0349
= 0.2646 × eA×0.0126 × eF×0.0711 × eV×(−0.0793) = 0.0432 × eA×0.0381 × eF×0.1954 × eV×(−0.2492) (11)
The correlative coefficients of the MRA model including adjusted R2 , sum of squares and mean square are shown in Table 3. From the table, it is evident that all adjusted R2 are close to 1 (100%) and the results show the input cutting parameters greatly affect the six output cutting qualities. The predicting data of the six cutting qualities computed from Eq. (11) are illustrated in Fig. 5. Also shown in the figure are the experimental data for the purpose of comparison. The MRA predicting errors of the six cutting qualities are shown in Fig. 10 and the average training error eave computed from Eq. (10) is 6.711%.
5.1.2.
Testing process of MRA models
After the training process, 14 sets of random combinations at various laser-cutting parameters relating to its output experimental qualities shown in Table 4 are used to test the MRA model. The experiment of every combination is repeated four times then an average value is obtained as experimental data. Fig. 6 shows that comparison between predicting data computed from Eq. (11) and experimental data for the MRA model. The predicting errors of the six output cutting qualities are shown in Fig. 10 and the average testing error eave calculated from Eq. (10) is 7.158%.
Table 3 – The relational parameters of MRA model Relational parameter
Adjusted R2 Sum of squares Mean square
Laser-cutting quality DLine
(e)
DLine
(c)
WHAZ
(e)
WHAZ
(c)
WLine
(e)
WLine
(c)
0.983 0.278 0.011
0.963 0.529 0.022
0.935 0.038 0.001
0.916 0.036 0.001
0.9 0.008 0.0004
0.93 0.007 0.0003
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Fig. 5 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the MRA training process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
5.2.
The predicting results of ANN model
5.2.1.
Training process of ANN model
The convergence of BP using the common GD algorithm is very low and results in a local minimum during training. In order to improve the disadvantages and decrease error, some algorithms have been proposed, such as QN algorithm (Chong and Zak, 1996), SCG algorithm (Moller, 1993) and LM algorithm. Among the improving algorithm, LM algorithm is the better algorithm compared other improving algorithms and obtains better performance, convergence and accurate predicting val-
ues and has been applied in many studies. In this section four algorithms including GD, QN, SCG and LM algorithms are used to simulate then choose optimal algorithm that has less predicting error to train ANN and obtain an ANN predicting model. Before training, the input cutting parameters need to scale that are called normalization and shown in Table 5. Based on the ANN structure introduced previously, some ANN parameters for four algorithms remaining to be specified before the training process runs. Four ANN parameters for GD algorithm name the number of neurons s(1) in the hidden layer, the learning rate, the momentum and the error goal.
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Table 4 – Extra 14 sets of laser-cutting parameters for the testing process Data set 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Current (A) 31 31 31 31 35 35 35 35 31 35 33 29 37 33
Frequency (kHz) 1 1.5 2.5 3 1 1.5 2.5 3 2 2 1 2.5 2 3.5
Cutting speed (mm/s) 1.5 2.5 1 1 1.5 2.5 2.5 3 3.5 0.5 0.5 3.5 1 2.5
The neuron number s(1) has an influence in the accuracy of the ANN model. The learning rate and momentum are important parameters that control the amount of change mode to the connection weight and related obviously (Hagan et al., 1996). In general large learning rate has larger correction weight and approaches target fast, but it produces over connection weight to lead to the oscillation and divergence. However, the appropriate value obtains fast convergence and avoids producing larger error (Singh et al., 2005). For QN algorithm two ANN parameters name the number of neurons s(1) in the hidden layer and the error goal (Chong and Zak, 1996). For SCG algorithm four ANN parameters are the number of neurons s(1) in the hidden layer, sigma that changed in weight for second derivation, lambda that regulated the indefiniteness of the Hessian and the error goal (Moller, 1993). Three ANN parameters for LM algorithm are the number of neurons s(1) in the hidden layer, the iteration factor ϑ defined in the sixth procedure of the aforementioned training process to determine the speed of convergence and the error goal (Dobrzanski et al., 2005; Singh et al., 2007; Bariani et al., 2004; Chelani et al., 2002; He et al., 2005). The error goal is related to both the convergence and accuracy of the ANN model in such a way that in general, larger error goal produces faster convergence but accompanies less accuracy for the ANN. There is no simple quantitative relation
Table 5 – The normalized inputs for input layer of the ANN for training process Laser-cutting parameters Current (A)
Original inputs 29 33 37
Normalized inputs for ANN 0 0.5 1
Frequency (kHz)
0.5 2 3.5
0 0.5 1
Cutting speed (mm/s)
0.5 2 3.5
0 0.5 1
to determine the optimal values for these ANN parameters. The only way to determine them is via the ANN training process to see which of the combination of these parameters will lead to the minimum predicting error. Fig. 7(a) shows the searching result of using GD algorithm over the parameter space with the number of neuron ranging from 3 to 7, the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }, the learning rate in the set of {0.001, 0.01 and 0.1} and the momentum in the set of {0.001, 0.01 and 0.1}. Fig. 7(b) shows the searching result by using QN algorithm over the parameter space with the number of neuron ranging from 3 to 7 and the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }. Fig. 7(c) shows the searching result by using SCG algorithm over the parameter space with the number of neuron ranging from 3 to 7, the sigma in the set of {10−2 , 10−3 , 10−4 and 10−5 }, the lambda in the set of {10−2 , 10−3 , 10−4 , 10−5 and 10−6 } and the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }. Fig. 7(d) shows the searching result of using LM algorithm over the parameter space with the number of neuron ranging from 3 to 7, with the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 } and the iteration factor ϑ ranging from 10 to 90 with increment 10. And the initial value of the factor k equals to 0.001. Other essential parameters in the ANN are set directly as: (1) initial weight equal to 0.01, (2) initial bias equal to 0.01 and (3) the number of training epochs equal to 1000. The training process was terminated by the error goal or the number of training epochs, depending on which is achieved first. The searching result in Fig. 7 reveals that the average predicting error of using LM algorithm achieves its minimum predicting error while the following ANN parameters are adopted: 5 neurons in the hidden layer, ϑ of 0.1 and the error goal of 10−4 . Then these parameters are used in the numerical computations involved in the ANN training process to obtain the mathematical model for predicting the six laser-cutting qualities. Finally, the ANN mathematical predicting model, i.e., ANN output layer a(2) = f(2) (n(2) ) = n(2) = W(2) a(1) + b(2) , that describes the relation between three laser-cutting parameters and six cutting qualities, is given by the following equation:
⎡
⎤ ⎡ 0.0001 0.1182 0.1730 −0.0004 ⎢ (c) ⎥ ⎢ ⎢ DLine ⎥ ⎢ −0.0001 0.3568 0.1415 0.0000 ⎢ ⎥ ⎢ W (e) ⎥ ⎢ ⎢ ⎢ Haz ⎥ ⎢ 0.7119 0.0068 0.0266 0.0577 a(2) =⎢ ⎥= (c) ⎥ ⎢ ⎢ WHaz 0.6113 −0.0098 0.0312 0.0533 ⎢ ⎥ ⎢ ⎢ (e) ⎥ ⎢ ⎣ 0.2384 0.0043 0.0362 0.0102 ⎣ WLine ⎦ (e)
DLine
0.2778
(c)
WLine
⎡
(1)
a1
⎤
⎡
0.3677
0.0059
0.0193
0.0192
0.2415
⎤
⎥ ⎥ ⎥ 0.0360 ⎥ ⎥ 0.0345 ⎥ ⎥ ⎥ 0.0402 ⎦ 0.2079
0.0284
⎤
⎢ ⎥ ⎢ (1) ⎥ ⎢ 0.1938 ⎥ ⎥ ⎢ a2 ⎥ ⎢ ⎢ (1) ⎥ ⎢ 0.1669 ⎥ ⎥ ⎥+⎢ ×⎢ a ⎢ 3 ⎥ ⎢ 0.1944 ⎥ ⎥ ⎢ (1) ⎥ ⎢ ⎥ ⎣ a4 ⎦ ⎢ ⎣ 0.3002 ⎦
(12.1)
(1)
a5
0.0649
where the outputs of the hidden layer a(1) = f(1) (n(1) ) = logsig(W(1) p + b(1) ) is given by the following equation:
278
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 270–283
Fig. 6 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the MRA testing process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
⎧⎡ ⎪ ⎪ ⎪ ⎢ ⎢ a(1) ⎥ ⎪ ⎪ ⎢ 2 ⎥ ⎨⎢ ⎢ ⎢ (1) ⎥ = ⎢ a3 ⎥ = logsig ⎢ ⎢ ⎢ (1) ⎥ ⎪ ⎪ ⎪ ⎣ ⎣ a4 ⎦ ⎪ ⎪ ⎩ ⎡
a(1)
(1)
a1
⎤
(1) a5
⎡ −3.512 ⎤⎫ ⎪ ⎪ ⎪ ⎢ −2.477 ⎥⎪ ⎢ ⎥⎬ ⎢ ⎥ + ⎢ 1.344 ⎥⎪ , ⎣ −10.16 ⎦⎪ ⎪ ⎪ ⎭ 20.67
1.513
2.822 −5.551
43.49
206.6
−125.0
81.87
71.76
−40.92
25.38
48.66
−38.61
1.860
54.80
−43.72
⎤ The predictions of the six cutting qualities using the above explicit formulas are illustrated in Fig. 8. Also shown in the figure are the experimental data for the purpose of comparison. The predicting errors of the six cutting qualities are depicted in Fig. 10 and the average training error eave computed from Eq. (10) is 0.822%.
⎥⎡A⎤ ⎥ ⎥⎣ ⎦ ⎥ F ⎥ ⎦ V
(12.2)
5.2.2.
Testing process of ANN model
After the training process, the generality ability of the ANN model must be tested by using 14 sets of random combinations of testing data shown in Table 4. The experiment of every
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 270–283
279
Fig. 7 – Comparisons of average predicting errors of ANN models using four algorithms. (a) GD algorithm using various combinations of number of neurons, learning rates, momentums and error goals, (b) QN algorithm using various combinations of number of neurons and error goals, (c) SCG algorithm using various combinations of number of neurons, sigma, lambda and error goals, and (d) LM algorithm using various combinations of number of neurons, the iterative factor (ϑ) and error goals.
combination is repeated four times then an average value is obtained as experimental data. Fig. 9 shows that comparison between predicting data computed from Eq. (11) and experimental data for the six cutting qualities of ANN model. The predicting errors of the six output cutting qualities are shown in Fig. 10. As expected, the predicting errors from the testing data are slightly larger than those from the training input data. Nevertheless, an average predicting error eave calculated from Eq. (10) is 1.512% for testing data. Therefore, Eq. (12) provides a reliable mathematical model in predicting six cutting qualities of a QFN package. From above results, the predicting model using ANN with LM algorithm matches the experimental data very well. Finally, the genetic algorithm is utilized further to find the optimal cutting parameters that lead to less HAZ width and fast cutting speed with the complete cutting.
6. The optimization of cutting quality using genetic algorithm The GA relating theories and calculations were firstly introduced by Holland (1975). The algorithm is a computational search scheme according to the mechanics of natural genetics and selection and is used to obtain optimal solutions (Goldberg, 1989). The operation procedures are described by following steps.
(1) Start with a randomly generated population of ‘n’ m-bit chromosomes. (2) Calculate the fitness function H of each of ‘x’ chromosomes in the population.
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Fig. 8 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the ANN training process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
(3) Use three operators that are selection, crossover and mutation to create ‘n’ offspring from current population. (4) The new population replaces the current population. (5) Go back to (2) and recomputed the fitness H then repeat step (3), (4) and (5) until the termination criterion is reached. In general situation, there is no effective way to determine the probabilities of crossover rate and mutation. In this study, the probability of crossover rate is set for 0.1 to 1 with increment 0.05 and the probability of mutation is set for 0.01 to 0.09 with increment 0.01. Other essential parameters for GA had
set: (1) roulette wheel selection, (2) double-point crossover, (3) elite of 1, (4) the size of generations of 100, (5) the size of populations of 50, and (6) the bit length of each variable of 10. Because the width of HAZ is significantly wider than the width of cutting line, the width of cutting line is not considered in the fitness function H. Then, the fitness function H design which is based on the quality consideration is described as follow: (1) A complete cutting, which is required for the QFN packages, must be transformed into a higher HD value.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 270–283
281
Fig. 9 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the ANN testing process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
Oppositely, an incomplete cutting is transformed into a lower HD value. The following Eqs. (13.1) and (13.2) define the equation HD of DLine , which can ensure that incomplete and complete cutting lead the weight coefficient to 0 and 1, respectively.
(e)
(e)
HD = 1,
if DLine ≥ 0.9 mm
(e) HD
if DLine < 0.9 mm
= 0,
(e)
(13.1)
(c)
HD = 1, (c)
HD = 0,
(c)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
(c)
(13.2)
(2) The smaller width of HAZ is desired and must be transformed into a higher HW value; oppositely, wider widths are transformed into a lower HW value. From the preliminary experiments, the measured width of HAZ varied from 0.1 mm to 0.8 mm for epoxy, and from 0.15 mm and 0.75 mm for copper circuit-pad bonding epoxy. The fol-
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Table 6 – Experimental and GA predicting qualities by using optimal cutting combination Data source
Laser-cutting quality (mm) (e)
GA Experiment data
DLine
DLine
(c)
WHAZ
(e)
0.9 0.90
0.9 0.90
0.2367 0.24
(c)
WHAZ 0.2508 0.25
Fig. 10 – Comparison of predicting errors of six laser-cutting qualities from MAR and ANN models. lowing Eqs. (13.3) and (13.4) are utilized to quantify the above principle and can ensure that maximum and minimum width of HAZ lead the weight coefficient to 0 and 1, respectively.
(e)
HW =
10 8 (e) − × WHAZ , 7 7
(e)
HW = 0,
(c)
HW =
5 5 (c) − × WHAZ , 4 3
(c)
(e)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
if
(e)
(13.4)
if DLine < 0.9 mm
(3) Furthermore, a faster cutting speed, which is preferred to reduce the cutting time, must be transformed into a higher HV value. Oppositely, a slower cutting speed is transformed into a lower HV value. The maximum and minimum of cutting speed V are 3.5 mm/s and 0.5 mm/s in this paper. Eqs. (13.5) and (13.6) is utilized to ensure that the slowest and the fastest cutting speed lead the weight coefficient to 0 and 1, respectively.
(e)
HV =
1 1 ×V− , 3 6
(e)
HV = 0,
(c)
HV =
1 1 ×V− , 3 6
(c)
(e)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
if
DLine ≥ 0.9 mm
(e)
(13.5)
(c) (c)
HV = 0,
(13.6)
if DLine < 0.9 mm
By integrating Eqs. (13.1)–(13.6), the fitness function H can be obtained as follow: (e)
(c)
(e)
(c)
(e)
Fig. 11 – A photograph of laser-cutting qualities by using optimal cutting parameters.
(c)
DLine ≥ 0.9 mm (c)
HW = 0,
(13.3)
(c)
H = HD + HD + HW + HW + HV + HV
7.
Conclusions
This paper reports the first application of an artificial neural work (ANN) for building a feasible predicting model for six laser-cutting qualities of QFN package. The back-propagation neural network with LM algorithm has been applied to construct a mathematical model wherein the six laser-cutting qualities are expressed as explicit nonlinear functions of the three laser control parameters A (current), F (frequency) and V (cutting speed). The established ANN model was trained from 27 sets of experimental data and tested by extra 14 sets of experimental data from a practical laser-cutting system. The average predicting errors are found to be 0.822% and 1.512% in the training and testing processes, respectively. The experimental results show that the built predicting model is reliable. Finally, a genetic algorithm is utilized to obtain the optimal cutting parameters that provide the best cutting qualities. The found combination is current of 29 A, frequency of 2.7 kHz and cutting speed of 3.49 mm/s. The GA is helpful to determine the best combination of laser-cutting parameters for optimizing a specified cutting quality.
(14)
From the GA simulation results, the optimal cutting combination is obtained with crossover rate of 0.5 and mutation of 0.03. The found combination is current of 29 A, frequency of 2.7 kHz and cutting speed of 3.49 mm/s. Table 6 shows that experimental and GA predicted cutting qualities. Fig. 11 shows an experimental photograph of laser-cutting qualities by using the optimal cutting combination.
Acknowledgements The authors would like to thank the Gallant Precision Machining Company Limited, Taiwan, for providing experimental materials and devices. The authors also want to thank Prof. Ciann-Dong Yang in National Cheng Kung University, Taiwan, for his support.
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journal homepage: www.elsevier.com/locate/jmatprotec
Optimal laser-cutting parameters for QFN packages by utilizing artificial neural networks and genetic algorithm Ming-Jong Tsai a,∗ , Chen-Hao Li a , Cheng-Che Chen a,b a
Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, No. 43, Keelung Road, Section 4, Taipei 10672, Taiwan b Gallant Precision Machining Company, Taipei, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, a multiple regression analysis (MRA) and an artificial neural network (ANN)
Received 30 July 2007
were employed to build a predicting model for cutting Quad Flat Non-lead (QFN) packages
Received in revised form
by using a Diode Pumped Solid State Laser (DPSSL) System. The predicting model includes
31 October 2007
three input variables of the current, the frequency and the cutting speed, and six cutting
Accepted 27 December 2007
qualities of depths of the cutting line, widths of heat affected zone (HAZ) and cutting line for epoxy and for copper-compounded epoxy. After the training process from 27 sets of training data including input data and its output qualities, the average training error is 0.822% by
Keywords:
using a back-propagation (BP) neural network with Levenberg–Marquardt (LM) algorithm,
QFN package
which leads to the best results. The testing accuracy is then verified with extra 14 sets
Laser cutting
of experimental data and the average predicting error is 1.512%. The results show that the
Multiple regression analysis (MRA)
ANN model has the predicting ability to estimate the laser-cutting qualities of QFN packages.
Artificial neural network (ANN)
Finally, a genetic algorithm (GA) is applied to find the optimal cutting parameters that lead
Levenberg–Marquardt (LM)
to least HAZ width and fast cutting speed with complete cutting. The optimal combination
algorithm
found is the current of 29 A, the frequency of 2.7 kHz and the cutting speed of 3.49 mm/s. The
Genetic algorithm (GA)
GA is helpful to determine the ideal laser-cutting parameters in order to meet the desired cutting qualities and to avoid unnecessary adjustments in the subsequent cutting process. © 2008 Elsevier B.V. All rights reserved.
1.
Introduction
A Quad Flat Non-lead (QFN) package is one of the main semiconductor packaging technologies. It consists of a plastic encapsulated package with a copper lead frame substrate. This packaging technology, which has many advantages, such as less size and weight, good electrical performance and high speed and frequency, has been applied widely in many products (Chen et al., 2003). In cutting QFN ICs, the conventional technology using diamond saw has the advantage of high-
∗
Corresponding author. Tel.: +886 2 27376286; fax: +886 2 27301265. E-mail address: [email protected] (M.-J. Tsai). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.12.138
speed cutting but tends to make the QFN ICs breakable and easy to crack. This is an area where laser cutting may offer a solution by providing a technique, which enables IC cutting to be carried out with less width of cutting line and less heat affected zone (HAZ) (Tsai et al., 2005; Li et al., 2007). Nowadays lasers are widely used in manufacturing and automotive industries, such as laser cutting, welding, marking, and drilling, in order to manufacture low cost, high quality products in short time. Many laser control parameters affecting laser-cutting qualities have been investigated,
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including the effect of laser-cutting speed on surface temperature (Rajendran and Pate, 1988), on surface quality (Neimeyer et al., 1993), and on surface roughness (Rajaram et al., 2003), and the effect of laser frequencies on HAZ (Kaebernick et al., 1998). In the meanwhile, many efforts have been made in the literature to determine the optimal values for these laser control parameters (Olsen, 1983; Steen and Kamalu, 1983; Hamoudi, 1996; Nagarajan, 2000). Nevertheless, the mechanism behind laser cutting is very complicated and process dependent, making it difficult to predict laser-cutting quality via analytical formulae whose validity often applies to limited range of processes and cutting conditions. Consequently, there is a need for a tool that will allow the evaluation of laser-cutting quality before the actual cutting process. Multiple regression analysis (MRA) is one method of statistics, which is utilized to build a mathematical model that describes the relation between input factors and output factors. This method is widely applied in many engineering, including the model of surface finish in electro-discharge machining (Petropoulos et al., 2004) and elastic properties of intact rocks (Karakus et al., 2005). Artificial neural networks (ANN) is an empirical modeling tool analogous to the behavior of biological neural structures and can identify highly complex relationships by using input–output data (Benardos and Vosniakos, 2002; Cheng and Lin, 2000; Dobrzanski et al., 2005; Singh et al., 2007; Bariani et al., 2004; Chelani et al., 2002; He et al., 2005). A common ANN model is established in a training process by using backpropagation (BP) with the gradient descent (GD) algorithm. To conquer the problem of slow convergence encountered by BP with GD, many improvements have been proposed, such as scaled conjugate gradient (SCG) algorithm, quasi-Newton (QN) algorithm and Levenberg–Marquardt (LM) algorithm. The algorithms improve common GD and produce less predicting error. In this paper above four algorithms (GD, SCG, QN and LM) are used to train the predicting model of laser-cutting qualities, respectively, and compare the predicting errors to obtain least errors and accurate predicting values. Genetic algorithm (GA) has been widely applied in many areas, such as control, prediction, forecasting, optimization, differentiation, vision, etc. The use of GA has become popular in recent years to optimize the process parameters and obtain optimal solutions in the engineering, including the optimization of the designing of chemical composition of high-speed steels (HSS) (Sitek and Dobrzanski, 2005), and the die shape in extrusion (Chung and Hwang, 1997), and optimal output
feedback controller (Badran and Al-Duwaish, 1999), and heat sources on a vertical wall with natural convection (Dias and Milanez, 2006), and dual-homing cell assignment problem of the two-level wireless ATM network (Din and Tseng, 2002) and controller parameters of heating ventilating and air conditioning (HVAC) (Huang and Lam, 1997). In this paper, the multiple regression analysis (MRA) and the artificial neural network are utilized to build a predicting model for the six laser-cutting qualities of the QFN packages, respectively. A total 41 sets of experimental data are used to train and verify the ANN model for predicting the cutting qualities. Finally, the genetic algorithm is utilized to find the optimal cutting parameters that lead to less HAZ width and fast cutting speed with the complete cutting.
2. Descriptions of QFN packages and laser-cutting system The internal structure, the cutting path and the microscopic dimensions of QFN packages are described in Fig. 1 (Li et al., 2007). A QFN package is a plastic encapsulated lead-frame based Chip Scale Package (CSP) with the lead pad on the bottom of the package to provide electrical interconnection with the printed circuit board (PCB) (Chen et al., 2003). Several QFN chips are packaged in an array form and must be separated into individual QFN chips before mounting on different printed-circuit boards. Because the whole size of a QFN package is small, the width of the tolerable cutting path between packaged ICs is also very restricted. For a 5 × 5 QFN package the maximum tolerant dimension of the cutting path is 0.26 mm and the thickness is 0.9 mm as shown in Fig. 1 (Li et al., 2007). The patch is made up of two materials, one is the epoxy with a thickness of 0.9 mm and the other is the copper circuit-pad bonding epoxy for which the dimensions of copper and epoxy are 0.24 mm and 0.66 mm, respectively. Fig. 2 sketches the six main parameters (e) governing the cutting quality, including the width (WHAZ ) of (e)
(e)
HAZ, the width (WLine ) and the depth (DLine ) of the cutting line (c) (c) for epoxy, and the three counterparts WHAZ , WLine
(c)
and DLine for
copper circuit-pad bonding epoxy (Li et al., 2007). (e)
(c)
The equipment for measuring the widths (WHAZ , WHAZ ) of (e)
(c)
HAZ and the widths (WLine , WLine ) of cutting line is an electron microscope with measurement accuracy of 1 m. The equip-
Fig. 1 – The cutting path and dimensions of a 5 × 5 QFN package.
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Fig. 2 – Geometrical illustration of six laser-cutting qualities of a QFN package.
(e)
(c)
ment for measuring the depths (DLine , DLine ) of the cutting line is an optical electron microscope with measurement accuracy of 1 m. Due to the irregularity of the widths of HAZ and cutting line, the values of widths along the cutting trajectory are indeed not constant. In order to obtain an accurate average measurement, a mean value of ten-point measurements is used. The depth measurement for the cutting line is taken at four fixed positions and a mean depth can be obtained. In this paper, the laser-cutting system shown in Fig. 3 consists of a diode pumped solid-state laser system (DPSSL; model: Rofin 100D, 1.064 m), a laser-cutting head, an X–Y moving table driven by two servomotors. The moving speed of the X–Y table is defined as the cutting speed in this paper. The current, the frequency and the cutting speed are three laser-cutting parameters adjusted by the controlling panel of the laser system.
3. Main laser control parameters and six cutting qualities In this section, the influences of laser power and cutting speed on the cutting qualities, including the widths and depths of cutting line and the widths of HAZ, will be discussed by using the aforementioned DPSSL system for cutting a QFN package.
3.1.
According to the previous reports (Tsai et al., 2005, 2006; Li et al., 2007), it was known that the dominant laser parameters affecting the cutting quality for a QFN package are the current A, the laser pulse frequency F and the cutting speed V, which are then taken as the main laser control parameters in this paper.
3.1.1.
The current
Using lower current provides lower peak-power for single pulse laser and reduces cutting heat so as to decrease the widths of cutting line and HAZ. However, applying lower current is difficult to cut off the materials effectively. On the contrary, a higher current can generate higher peak-power and release more cutting heat and thus increase the cutting line and HAZ widths. Nevertheless, the cutting process may be more efficient by adopting higher current.
3.1.2.
The frequency
The lower frequency usually leads to higher peak-power of single pulse and gives higher cutting ability. However, the lower frequency also accompanies low-level pulse overlapping and non-continuous power density within a unit length. Therefore, the lower frequency laser cutting tends to produce a noncontinuous cutting line and to generate burn spots along the cutting path. On the contrary, higher frequency cutting has relatively lower peak-power for single pulse and leads to less cutting ability, but it provides higher level pulse overlapping to yield continuous power density within a unit length.
3.1.3.
Fig. 3 – The laser-cutting system for QFN packages.
Main laser control parameters
The cutting speed
Applying lower speed cutting process often accompanies higher level overlapping and continuous power density per unit length and hence tends to cut completely; but it produces more heat and wider HAZ and cutting line. On the contrary, a higher speed cutting process results in narrow HAZ and cutting line at the risk of producing incomplete cutting. In fact, the average output power (Watt) of a laser system is a major factor for providing the required cutting energy. The available output power Y that is measured by using a DPSSL
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system is produced at various combinations of the current A and the laser pulse frequency F. The single laser pulse power K (Joule) is expressed by the following equation (Thawari et al., 2005):
y3j = k30j + Ak31j + Fk32j + V k33j
F , V
Yp = K × S =
Y , V
(2)
where V is the cutting speed and S is also known as pulse overlapping, which is an indication of the smoothness and the continuity of laser cutting. The variations of the power per unit length Yp at various combinations of current A, frequency F and cutting speed V are discussed and illustrated in our previous paper (Li et al., 2007). In order to build the mathematical relations between the three laser control parameter and six cutting qualities, a column vector p containing three input cutting parameters is defined as follow: T
T
p = [A F V] = [current, frequency, cutting speed]
3.2.
y4j = k40j × Ak41j × Fk42j × V k43j
(e)
(c)
t = D(e) Line
(c)
DLine
(e)
WHAZ
(c)
WHAZ
(e)
WLine
(c)
WLine
T
y5j = k50j + eA×k51j + eF×k52j + eV×k53j
y6j = k60j × eA×k61j × eF×k62j × eV×k63j
(4.6)
where yij is jth output cutting quality of ith model (i = 1–6
type
(e)
and
(c)
(e)
(c)
(e)
(c)
j = 1 − 6 = DLine , DLine , WHAZ , WHAZ ,
WLine , WLine ), A, F, V are laser input cutting parameters and kidj is dth regression coefficient at jth output cutting quality of ith model (d = 0–3). And all regression coefficients are estimated for MRA model. In this paper, the analyses were implemented by using Statistics Package for Social Science (SPSS) software package.
4.2.
The ANN model
The structure of the ANN adopted in this paper includes three layers: an input layer, a hidden layer and an output layer, as illustrated in Fig. 4. The input layer receives three external input parameters (A, F, V) from the laser-cutting system and the output layer provides the outputs of the six laser-cutting qualities as follow:
(3.2) (2)
(2)
(2)
(2)
(2)
(2)
(e)
(c)
(e)
(c)
(f1 , f2 , f3 , f4 , f5 , f6 ) = (DLine , DLine , WHAZ , WHAZ , (e)
(c)
WLine , WLine ).
In this section, the relations between laser control parameters for cutting a QFN package and six output cutting qualities were modeled by using MRA and ANN methods.
The MRA models
For MRA modeling, six different multiple regression models that include linear, power and exponential relations are adopted for predictions of six output cutting qualities of a QFN package, respectively. The six MRA models are described as follows: (1) Model 1: First type linear relation (4.1)
(5)
To accommodate the required input and output numbers, three and six neurons must be used in the input and output layers, respectively, i.e., s(0) = 3 and s(2) = 6. The neuron number s(1) in the hidden layer is free to choose, but a best s(1) will be determined later in the next section. The transfer function used in the hidden layer is in the form of log-sigmoid function as follow:
⎡
⎤ ⎡ 1 (1) ⎢ 1 + exp(−n ⎥ 1,q ) (1) (1) f2 (n2,q ) ⎥ ⎢ ⎢ .. ⎥ ⎢ ⎥=⎢ . .. ⎥ . ⎦ ⎢ 1 ⎣ (1)
⎤
(1)
f1 (n1,q )
⎢ ⎢ (1) (1) ⎢ f (nq )=⎢ ⎢ ⎣ f
y1j = k10j + k11j A + k12j F + k13j V
(4.5)
(6) Model 6: Second type exponential relation
4. The MRA and ANN models for laser-cutting qualities
4.1.
(4.4)
(5) Model 5: First type exponential relation
Six output cutting qualities
of cutting line (WLine , WLine ) were described in Tsai et al. (2006), Cenna and Mathew (2005) and some experimental results were reported in Li et al. (2007). A column vector t containing six output cutting qualities of a QFN package is defined as follow:
(4.3)
(4) Model 4: Second type power relation
(3.1)
The theoretical models describing their relationships between the average output power Y, the cutting speed V and the six output qualities including the depth of cutting line (e) (c) (e) (c) (DLine , DLine ), the widths of HAZ (WHAZ , WHAZ ) and the widths
(4.2)
(1)
The other two measures of power dominating the cutting quality include the number of laser pulses per unit length S and the power per unit length Yp (Joule/mm), which can be obtained by following equation: S=
y2j = k21j A + k22j F + k23j V (3) Model 3: First type power relation
Y . F
K=
(2) Model 2: Second type linear relation
(1)
s(1)
(1)
(n
s(1) ,q
)
(1)
1 + exp(−n
s(1) ,q
⎥ ⎥ ⎥ ⎥ , q=1, 2, . . . , 27, ⎥ ⎥ ⎦ ) (6)
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Fig. 4 – The ANN structure for predicting six laser-cutting qualities of a QFN package.
(1)
(0)
where nq = W(1) aq + b(1) is the net input of the hidden layer at qth data indicating input parameters relating to its output (2) qualities (Hagan et al., 1996). As to the transfer functions fj for the output neurons, the linear relation is employed for the last four outputs: (2) (2) fj (nj,q )
=
(2) nj,q ,
j = 3, 4, 5, 6.
(7) (2)
(e)
(2)
(c)
The first two cutting qualities f1 = DLine and f2 = DLine need separate consideration. Because the thickness of QFN package in the present case is 0.9 mm, a cutting depth greater than 0.9 mm means a complete cutting. Accordingly, the outputs of the first and second neurons regarding cutting depth are defined as follow:
(2)
(2)
fj (nj,q ) =
⎧ 0, ⎪ ⎪ ⎨
(2)
n ,
(2)
nj,q ≤ 0 0
(2)
< 0.9 ,
j,q j,q ⎪ ⎪ ⎩ 0.9, n(2) ≥ 0.9 j,q
j = 1, 2.
(8)
With the specified transfer functions f(1) and f(2) , the out(2) puts aq of the ANN regarding the cutting quality predictions can be determined iteratively from the reference (Hagan et al., 1996) along with the iterations of the weights W(1) and W(2) , and biases b(1) and b(2) . The ANN is trained by 27 sets of inputs listed in Table 1 and by the associated measurement outputs recorded in Li et al. (2007). By learning the patterns of these inputs and outputs, the weights and biases are adjusted iteratively during the training procedures (Hagan et al., 1996). At the end of training process, the ANN predicting errors for each
Table 1 – Twenty-seven sets of laser-cutting parameters for the training process Data set 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
Current (A)
Frequency (kHz)
29 29 29 29 29 29 29 29 29 33 33 33 33 33 33 33 33 33 37 37 37 37 37 37 37 37 37
0.5 0.5 0.5 2 2 2 3.5 3.5 3.5 0.5 0.5 0.5 2 2 2 3.5 3.5 3.5 0.5 0.5 0.5 2 2 2 3.5 3.5 3.5
Cutting speed (mm/s) 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5 0.5 2 3.5
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Table 2 – The predicting errors of six MRA models using 27 sets of training data Model type
Laser-cutting quality
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
DLine
(e)
DLine
(c)
9.603 6.921* 10.089 11.228 10.836 11.524
18.316 7.602* 19.068 23.867 21.477 25.27
(e)
(c)
WLine
WLine
14.403 13.968 13.564 13.159 21.355 7.482*
3.9225 6.528 4.457 6.448 8.875 3.132*
13.949 13.734 12.984 16.449 26.537 7.713*
WHAZ
(e)
WHAZ
17.021 16.359 16.388 15.532 25.589 7.416*
(c)
(*) Represent the least predicting error; unit: %.
cutting quality are defined as follows: |tj,q − yj,q |
ej,q (%) =
tj,q
× 100%,
j = 1–6
1 ej,q 27
regression equations of six cutting qualities can be estimated as follows: (9.1)
27
ej (%) =
(9.2)
q=1
where ej,q is jth output cutting quality at qth data and is an element of the error vector v defined in the reference (Hagan et al., 1996). Therefore, the average predicting error of the six cutting qualities can be obtained by
eave (%) =
5.
6 1
6
ej .
(10)
j=1
Results and discussions
According to the previous experiments (Li et al., 2007), the lower and upper bounds for every cutting parameter are determined, i.e., current between 29 A and 37 A, frequency between 0.5 kHz and 3.5 kHz, and cutting speed between 0.5 mm/s and 3.5 mm/s. With the established ranges of cutting parameters, the maximum, the minimum and the mean are chosen as the three levels for each cutting parameter. That is to say, the three levels for current are 29 A, 33 A and 37 A; frequency are 0.5 kHz, 2 kHz and 3.5 kHz, and cutting speed are 0.5 mm/s, 2 mm/s and 3.5 mm/s. Table 1 summarizes the laser-cutting parameters and their levels. From the table, 27 sets of various combinations of input cutting parameters are chosen to experiment by using a DPSSL then every combination obtains six cutting qualities. By the way, the experiment of each combination is repeated four times to give an average value. The 27 sets combinations that present three input cutting parameters and six cutting qualities are adopted as training data.
5.1.
The predicting results of MRA models
5.1.1.
Training process of MRA models
A total of 27 sets of training data are employed to train MRA model. Table 2 shows that the predicting errors e1 , e2 , . . ., e6 of the six cutting qualities computed from Eq. (9) at MRA model of six types. From the table, it is observed (e) (c) that the predicting errors of DLine , DLine using model 2 and (e)
(c)
(e)
(c)
WHAZ , WHAZ , WLine , WLine using model 6 are least. The multiple
⎧ (e) DLine = (0.0217 × A) + (0.0915 × F) + ((−0.0529) × V) ⎪ ⎪ ⎪ ⎪ D(c) = (0.0188 × A) + (0.0141 × F) + ((−0.0859) × V) ⎪ ⎪ ⎪ ⎪ Line (e) ⎨ A×0.0414 F×0.2148 V×(−0.278) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
WHAZ = 0.0856 × e
×e
×e
(c) WHAZ (e) WLine (c) WLine
× eF×0.1775
× eV×(−0.2302)
=
0.1098 × eA×0.0349
= 0.2646 × eA×0.0126 × eF×0.0711 × eV×(−0.0793) = 0.0432 × eA×0.0381 × eF×0.1954 × eV×(−0.2492) (11)
The correlative coefficients of the MRA model including adjusted R2 , sum of squares and mean square are shown in Table 3. From the table, it is evident that all adjusted R2 are close to 1 (100%) and the results show the input cutting parameters greatly affect the six output cutting qualities. The predicting data of the six cutting qualities computed from Eq. (11) are illustrated in Fig. 5. Also shown in the figure are the experimental data for the purpose of comparison. The MRA predicting errors of the six cutting qualities are shown in Fig. 10 and the average training error eave computed from Eq. (10) is 6.711%.
5.1.2.
Testing process of MRA models
After the training process, 14 sets of random combinations at various laser-cutting parameters relating to its output experimental qualities shown in Table 4 are used to test the MRA model. The experiment of every combination is repeated four times then an average value is obtained as experimental data. Fig. 6 shows that comparison between predicting data computed from Eq. (11) and experimental data for the MRA model. The predicting errors of the six output cutting qualities are shown in Fig. 10 and the average testing error eave calculated from Eq. (10) is 7.158%.
Table 3 – The relational parameters of MRA model Relational parameter
Adjusted R2 Sum of squares Mean square
Laser-cutting quality DLine
(e)
DLine
(c)
WHAZ
(e)
WHAZ
(c)
WLine
(e)
WLine
(c)
0.983 0.278 0.011
0.963 0.529 0.022
0.935 0.038 0.001
0.916 0.036 0.001
0.9 0.008 0.0004
0.93 0.007 0.0003
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Fig. 5 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the MRA training process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
5.2.
The predicting results of ANN model
5.2.1.
Training process of ANN model
The convergence of BP using the common GD algorithm is very low and results in a local minimum during training. In order to improve the disadvantages and decrease error, some algorithms have been proposed, such as QN algorithm (Chong and Zak, 1996), SCG algorithm (Moller, 1993) and LM algorithm. Among the improving algorithm, LM algorithm is the better algorithm compared other improving algorithms and obtains better performance, convergence and accurate predicting val-
ues and has been applied in many studies. In this section four algorithms including GD, QN, SCG and LM algorithms are used to simulate then choose optimal algorithm that has less predicting error to train ANN and obtain an ANN predicting model. Before training, the input cutting parameters need to scale that are called normalization and shown in Table 5. Based on the ANN structure introduced previously, some ANN parameters for four algorithms remaining to be specified before the training process runs. Four ANN parameters for GD algorithm name the number of neurons s(1) in the hidden layer, the learning rate, the momentum and the error goal.
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Table 4 – Extra 14 sets of laser-cutting parameters for the testing process Data set 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Current (A) 31 31 31 31 35 35 35 35 31 35 33 29 37 33
Frequency (kHz) 1 1.5 2.5 3 1 1.5 2.5 3 2 2 1 2.5 2 3.5
Cutting speed (mm/s) 1.5 2.5 1 1 1.5 2.5 2.5 3 3.5 0.5 0.5 3.5 1 2.5
The neuron number s(1) has an influence in the accuracy of the ANN model. The learning rate and momentum are important parameters that control the amount of change mode to the connection weight and related obviously (Hagan et al., 1996). In general large learning rate has larger correction weight and approaches target fast, but it produces over connection weight to lead to the oscillation and divergence. However, the appropriate value obtains fast convergence and avoids producing larger error (Singh et al., 2005). For QN algorithm two ANN parameters name the number of neurons s(1) in the hidden layer and the error goal (Chong and Zak, 1996). For SCG algorithm four ANN parameters are the number of neurons s(1) in the hidden layer, sigma that changed in weight for second derivation, lambda that regulated the indefiniteness of the Hessian and the error goal (Moller, 1993). Three ANN parameters for LM algorithm are the number of neurons s(1) in the hidden layer, the iteration factor ϑ defined in the sixth procedure of the aforementioned training process to determine the speed of convergence and the error goal (Dobrzanski et al., 2005; Singh et al., 2007; Bariani et al., 2004; Chelani et al., 2002; He et al., 2005). The error goal is related to both the convergence and accuracy of the ANN model in such a way that in general, larger error goal produces faster convergence but accompanies less accuracy for the ANN. There is no simple quantitative relation
Table 5 – The normalized inputs for input layer of the ANN for training process Laser-cutting parameters Current (A)
Original inputs 29 33 37
Normalized inputs for ANN 0 0.5 1
Frequency (kHz)
0.5 2 3.5
0 0.5 1
Cutting speed (mm/s)
0.5 2 3.5
0 0.5 1
to determine the optimal values for these ANN parameters. The only way to determine them is via the ANN training process to see which of the combination of these parameters will lead to the minimum predicting error. Fig. 7(a) shows the searching result of using GD algorithm over the parameter space with the number of neuron ranging from 3 to 7, the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }, the learning rate in the set of {0.001, 0.01 and 0.1} and the momentum in the set of {0.001, 0.01 and 0.1}. Fig. 7(b) shows the searching result by using QN algorithm over the parameter space with the number of neuron ranging from 3 to 7 and the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }. Fig. 7(c) shows the searching result by using SCG algorithm over the parameter space with the number of neuron ranging from 3 to 7, the sigma in the set of {10−2 , 10−3 , 10−4 and 10−5 }, the lambda in the set of {10−2 , 10−3 , 10−4 , 10−5 and 10−6 } and the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 }. Fig. 7(d) shows the searching result of using LM algorithm over the parameter space with the number of neuron ranging from 3 to 7, with the error goal in the set of {10−1 , 10−2 , 10−3 and 10−4 } and the iteration factor ϑ ranging from 10 to 90 with increment 10. And the initial value of the factor k equals to 0.001. Other essential parameters in the ANN are set directly as: (1) initial weight equal to 0.01, (2) initial bias equal to 0.01 and (3) the number of training epochs equal to 1000. The training process was terminated by the error goal or the number of training epochs, depending on which is achieved first. The searching result in Fig. 7 reveals that the average predicting error of using LM algorithm achieves its minimum predicting error while the following ANN parameters are adopted: 5 neurons in the hidden layer, ϑ of 0.1 and the error goal of 10−4 . Then these parameters are used in the numerical computations involved in the ANN training process to obtain the mathematical model for predicting the six laser-cutting qualities. Finally, the ANN mathematical predicting model, i.e., ANN output layer a(2) = f(2) (n(2) ) = n(2) = W(2) a(1) + b(2) , that describes the relation between three laser-cutting parameters and six cutting qualities, is given by the following equation:
⎡
⎤ ⎡ 0.0001 0.1182 0.1730 −0.0004 ⎢ (c) ⎥ ⎢ ⎢ DLine ⎥ ⎢ −0.0001 0.3568 0.1415 0.0000 ⎢ ⎥ ⎢ W (e) ⎥ ⎢ ⎢ ⎢ Haz ⎥ ⎢ 0.7119 0.0068 0.0266 0.0577 a(2) =⎢ ⎥= (c) ⎥ ⎢ ⎢ WHaz 0.6113 −0.0098 0.0312 0.0533 ⎢ ⎥ ⎢ ⎢ (e) ⎥ ⎢ ⎣ 0.2384 0.0043 0.0362 0.0102 ⎣ WLine ⎦ (e)
DLine
0.2778
(c)
WLine
⎡
(1)
a1
⎤
⎡
0.3677
0.0059
0.0193
0.0192
0.2415
⎤
⎥ ⎥ ⎥ 0.0360 ⎥ ⎥ 0.0345 ⎥ ⎥ ⎥ 0.0402 ⎦ 0.2079
0.0284
⎤
⎢ ⎥ ⎢ (1) ⎥ ⎢ 0.1938 ⎥ ⎥ ⎢ a2 ⎥ ⎢ ⎢ (1) ⎥ ⎢ 0.1669 ⎥ ⎥ ⎥+⎢ ×⎢ a ⎢ 3 ⎥ ⎢ 0.1944 ⎥ ⎥ ⎢ (1) ⎥ ⎢ ⎥ ⎣ a4 ⎦ ⎢ ⎣ 0.3002 ⎦
(12.1)
(1)
a5
0.0649
where the outputs of the hidden layer a(1) = f(1) (n(1) ) = logsig(W(1) p + b(1) ) is given by the following equation:
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Fig. 6 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the MRA testing process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
⎧⎡ ⎪ ⎪ ⎪ ⎢ ⎢ a(1) ⎥ ⎪ ⎪ ⎢ 2 ⎥ ⎨⎢ ⎢ ⎢ (1) ⎥ = ⎢ a3 ⎥ = logsig ⎢ ⎢ ⎢ (1) ⎥ ⎪ ⎪ ⎪ ⎣ ⎣ a4 ⎦ ⎪ ⎪ ⎩ ⎡
a(1)
(1)
a1
⎤
(1) a5
⎡ −3.512 ⎤⎫ ⎪ ⎪ ⎪ ⎢ −2.477 ⎥⎪ ⎢ ⎥⎬ ⎢ ⎥ + ⎢ 1.344 ⎥⎪ , ⎣ −10.16 ⎦⎪ ⎪ ⎪ ⎭ 20.67
1.513
2.822 −5.551
43.49
206.6
−125.0
81.87
71.76
−40.92
25.38
48.66
−38.61
1.860
54.80
−43.72
⎤ The predictions of the six cutting qualities using the above explicit formulas are illustrated in Fig. 8. Also shown in the figure are the experimental data for the purpose of comparison. The predicting errors of the six cutting qualities are depicted in Fig. 10 and the average training error eave computed from Eq. (10) is 0.822%.
⎥⎡A⎤ ⎥ ⎥⎣ ⎦ ⎥ F ⎥ ⎦ V
(12.2)
5.2.2.
Testing process of ANN model
After the training process, the generality ability of the ANN model must be tested by using 14 sets of random combinations of testing data shown in Table 4. The experiment of every
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279
Fig. 7 – Comparisons of average predicting errors of ANN models using four algorithms. (a) GD algorithm using various combinations of number of neurons, learning rates, momentums and error goals, (b) QN algorithm using various combinations of number of neurons and error goals, (c) SCG algorithm using various combinations of number of neurons, sigma, lambda and error goals, and (d) LM algorithm using various combinations of number of neurons, the iterative factor (ϑ) and error goals.
combination is repeated four times then an average value is obtained as experimental data. Fig. 9 shows that comparison between predicting data computed from Eq. (11) and experimental data for the six cutting qualities of ANN model. The predicting errors of the six output cutting qualities are shown in Fig. 10. As expected, the predicting errors from the testing data are slightly larger than those from the training input data. Nevertheless, an average predicting error eave calculated from Eq. (10) is 1.512% for testing data. Therefore, Eq. (12) provides a reliable mathematical model in predicting six cutting qualities of a QFN package. From above results, the predicting model using ANN with LM algorithm matches the experimental data very well. Finally, the genetic algorithm is utilized further to find the optimal cutting parameters that lead to less HAZ width and fast cutting speed with the complete cutting.
6. The optimization of cutting quality using genetic algorithm The GA relating theories and calculations were firstly introduced by Holland (1975). The algorithm is a computational search scheme according to the mechanics of natural genetics and selection and is used to obtain optimal solutions (Goldberg, 1989). The operation procedures are described by following steps.
(1) Start with a randomly generated population of ‘n’ m-bit chromosomes. (2) Calculate the fitness function H of each of ‘x’ chromosomes in the population.
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Fig. 8 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the ANN training process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
(3) Use three operators that are selection, crossover and mutation to create ‘n’ offspring from current population. (4) The new population replaces the current population. (5) Go back to (2) and recomputed the fitness H then repeat step (3), (4) and (5) until the termination criterion is reached. In general situation, there is no effective way to determine the probabilities of crossover rate and mutation. In this study, the probability of crossover rate is set for 0.1 to 1 with increment 0.05 and the probability of mutation is set for 0.01 to 0.09 with increment 0.01. Other essential parameters for GA had
set: (1) roulette wheel selection, (2) double-point crossover, (3) elite of 1, (4) the size of generations of 100, (5) the size of populations of 50, and (6) the bit length of each variable of 10. Because the width of HAZ is significantly wider than the width of cutting line, the width of cutting line is not considered in the fitness function H. Then, the fitness function H design which is based on the quality consideration is described as follow: (1) A complete cutting, which is required for the QFN packages, must be transformed into a higher HD value.
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281
Fig. 9 – Comparisons of six laser-cutting qualities between the experimental data and the prediction data from the ANN testing process. (a) Depth of cutting line for epoxy, (b) depth of cutting line for epoxy + Cu, (c) width of HAZ for epoxy (d) Width of HAZ for epoxy + Cu, (e) width of cutting line for epoxy, and (f) width of cutting line for epoxy + Cu.
Oppositely, an incomplete cutting is transformed into a lower HD value. The following Eqs. (13.1) and (13.2) define the equation HD of DLine , which can ensure that incomplete and complete cutting lead the weight coefficient to 0 and 1, respectively.
(e)
(e)
HD = 1,
if DLine ≥ 0.9 mm
(e) HD
if DLine < 0.9 mm
= 0,
(e)
(13.1)
(c)
HD = 1, (c)
HD = 0,
(c)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
(c)
(13.2)
(2) The smaller width of HAZ is desired and must be transformed into a higher HW value; oppositely, wider widths are transformed into a lower HW value. From the preliminary experiments, the measured width of HAZ varied from 0.1 mm to 0.8 mm for epoxy, and from 0.15 mm and 0.75 mm for copper circuit-pad bonding epoxy. The fol-
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Table 6 – Experimental and GA predicting qualities by using optimal cutting combination Data source
Laser-cutting quality (mm) (e)
GA Experiment data
DLine
DLine
(c)
WHAZ
(e)
0.9 0.90
0.9 0.90
0.2367 0.24
(c)
WHAZ 0.2508 0.25
Fig. 10 – Comparison of predicting errors of six laser-cutting qualities from MAR and ANN models. lowing Eqs. (13.3) and (13.4) are utilized to quantify the above principle and can ensure that maximum and minimum width of HAZ lead the weight coefficient to 0 and 1, respectively.
(e)
HW =
10 8 (e) − × WHAZ , 7 7
(e)
HW = 0,
(c)
HW =
5 5 (c) − × WHAZ , 4 3
(c)
(e)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
if
(e)
(13.4)
if DLine < 0.9 mm
(3) Furthermore, a faster cutting speed, which is preferred to reduce the cutting time, must be transformed into a higher HV value. Oppositely, a slower cutting speed is transformed into a lower HV value. The maximum and minimum of cutting speed V are 3.5 mm/s and 0.5 mm/s in this paper. Eqs. (13.5) and (13.6) is utilized to ensure that the slowest and the fastest cutting speed lead the weight coefficient to 0 and 1, respectively.
(e)
HV =
1 1 ×V− , 3 6
(e)
HV = 0,
(c)
HV =
1 1 ×V− , 3 6
(c)
(e)
if
DLine ≥ 0.9 mm
if
DLine < 0.9 mm
if
DLine ≥ 0.9 mm
(e)
(13.5)
(c) (c)
HV = 0,
(13.6)
if DLine < 0.9 mm
By integrating Eqs. (13.1)–(13.6), the fitness function H can be obtained as follow: (e)
(c)
(e)
(c)
(e)
Fig. 11 – A photograph of laser-cutting qualities by using optimal cutting parameters.
(c)
DLine ≥ 0.9 mm (c)
HW = 0,
(13.3)
(c)
H = HD + HD + HW + HW + HV + HV
7.
Conclusions
This paper reports the first application of an artificial neural work (ANN) for building a feasible predicting model for six laser-cutting qualities of QFN package. The back-propagation neural network with LM algorithm has been applied to construct a mathematical model wherein the six laser-cutting qualities are expressed as explicit nonlinear functions of the three laser control parameters A (current), F (frequency) and V (cutting speed). The established ANN model was trained from 27 sets of experimental data and tested by extra 14 sets of experimental data from a practical laser-cutting system. The average predicting errors are found to be 0.822% and 1.512% in the training and testing processes, respectively. The experimental results show that the built predicting model is reliable. Finally, a genetic algorithm is utilized to obtain the optimal cutting parameters that provide the best cutting qualities. The found combination is current of 29 A, frequency of 2.7 kHz and cutting speed of 3.49 mm/s. The GA is helpful to determine the best combination of laser-cutting parameters for optimizing a specified cutting quality.
(14)
From the GA simulation results, the optimal cutting combination is obtained with crossover rate of 0.5 and mutation of 0.03. The found combination is current of 29 A, frequency of 2.7 kHz and cutting speed of 3.49 mm/s. Table 6 shows that experimental and GA predicted cutting qualities. Fig. 11 shows an experimental photograph of laser-cutting qualities by using the optimal cutting combination.
Acknowledgements The authors would like to thank the Gallant Precision Machining Company Limited, Taiwan, for providing experimental materials and devices. The authors also want to thank Prof. Ciann-Dong Yang in National Cheng Kung University, Taiwan, for his support.
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