Journal of ELECTRONIC MATERIALS, Vol. 40, No. 7, 2011
DOI: 10.1007/s11664-011-1631-z Ó 2011 TMS
Modeling the Power Output of Piezoelectric Energy Harvesters MAHMOUD AL AHMAD1 and H.N. ALSHAREEF1,2 1.—Materials Science and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia. 2.—e-mail:
[email protected]
Design of experiments and multiphysics analyses were used to develop a parametric model for a d33-based cantilever. The analysis revealed that the most significant parameters influencing the resonant frequency are the supporting layer thickness, piezoelectric layer thickness, and cantilever length. On the other hand, the most important factors affecting the charge output are the piezoelectric thickness and the interdigitated electrode dimensions. The accuracy of the developed model was confirmed and showed less than 1% estimation error compared with a commercial simulation package. To estimate the power delivered to a load, the electric current output from the piezoelectric generator was calculated. A circuit model was built and used to estimate the power delivered to a load, which compared favorably to experimentally published power data on actual cantilevers of similar dimensions. Key words: d33, energy harvesting, interdigitated electrode, parametric modeling, piezoelectric power generator
INTRODUCTION The search for alternative energy sources continues on many fronts. In particular, there has been a recent increase in harvesting energy from nontraditional alternative energy sources for selfpowered sensors that require relatively low power for operation.1 Examples of such applications include sensors that need to be placed inside buildings or embedded inside structures, implantable medical devices, and sensors in remote or hidden areas.2–5 The direct piezoelectric effect describes the conversion of mechanical strain into electrical current or voltage.6 This strain can come from many different sources such as human motion, lowfrequency seismic vibrations, and acoustic noise. Piezoelectric-based cantilevers are attractive for power harvesting from such sources because of their ability to withstand large amounts of strain due to their high deflection and low stiffness characteristics. There are two commonly used operation modes in piezoelectric cantilever devices (d31 and d33). In the d31 mode, the piezoelectric material is sandwiched between top and bottom electrodes, while (Received January 7, 2011; accepted March 19, 2011; published online April 30, 2011)
the use of the d33 mode eliminates the need for a bottom electrode by employing an interdigitated (IDT) top electrode configuration.7,8 It has been shown that d33 mode design gives higher open-circuit voltage than d31-type transducers for similar beam dimensions.9 The electric field in the case of the interdigitated electrode structure is parallel to the strain induced in the cantilever, which results in the d33 mode of operation.10 Several studies have reported on the use of d33 piezoelectric energy harvesters.11–14 The reported literature does not explicitly address which material and electrode configuration parameters are most significant for maximizing the energy harvested from vibrations. It would be useful to develop an analytical model that can quantify and predict the charge output of piezoelectric cantilevers and to predict the power delivered to a load as a function of such parameters. Therefore, the goal of this work is to develop an analytical parametric model that can be used to estimate the resonant frequency and charge output of d33 cantilevers given certain input parameters such as cantilever geometry, material, and electrode configuration. Once the charge output is calculated, a circuit model is used to estimate the power delivered to a load. The developed analytical model, which is based on the design of experiments 1477
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(DoE) method,15 can be used to optimize the IDT structure design, material dimensions, and cantilever performance without the need for expensive software packages or lengthy simulation times.
(a)
PIEZOELECTRIC MATERIALS Piezoelectric energy harvesting makes use of the direct piezoelectric effect. This occurs by trapping the cyclic mechanical stress from an available source and then converting the mechanical energy (Wmech) into electrical energy (Wel).16 An electrical charge is accumulated at the electrodes of the piezoelectric as a result of the distortion of the lattice.17,18 Piezoelectric cantilevers vibrate with the greatest amplitude at their resonant frequency, where the resonant frequency, fn , of a clamped-free cantilever without a proof mass can be expressed as19,20 rffiffiffiffiffiffi v2 1 EI ; (1) fn ¼ n 2 m 2p L where vn is the nth mode eigenvalue, L is the length of the cantilever, E is the modulus of elasticity, I is the area moment of inertia about the neutral axis, and m is the mass per unit length of the cantilever beam. When a cantilever structure (Fig. 1) is loaded with a force (F), electrical energy (Wel) is stored in the piezoelectric element, given by Wel ¼ 1=2C U 2 ;
(2)
where C is the capacitance and U is the generated voltage. The mechanical energy due to the deformation can be expressed as Wmech
¼ 1=2T 2 sD 33 vol;
(3)
where T is the mechanical strain, vol is the volume, and sD 33 is the elastic constant. Therefore, the electrical energy Wel and mechanical energy Wmech are related by Wel ¼
k233
Wmech ;
(4)
where k33 is the electromechanical coupling coefficient. Hence, the mechanical to electrical energy conversion depends on the piezoelectric properties, volume, and both the magnitude and the direction of the mechanical excitation. It is established that the maximum energy output is achieved at the resonant frequency of the cantilever, whereas the maximum power output is obtained when the piezoelectric’s impedance matches the load.21 The strain energy (SE) stored in the cantilever structure can be expressed as 1 SE ¼ ðFEff dÞ; 2
(5)
where d is the deflection and FEff is the effective force, which is equal to the vector sum of the
(b)
(c)
Fig. 1. Illustration of a d33-mode piezoelectric device: (a) device cross-section, (b) interdigitated electrode topology, and (c) the mechanical stress and electric field directions.
external and inertial forces and can be calculated as SE : (6) FEff ¼ 2 d The relationship between a force applied to a piezoelectric element and the charge produced can therefore be expressed as Q ¼ ðd33 FEff Þ ¼ 2ðd33 SE=dÞ:
(7)
The produced charge is directly proportional to the piezoelectric coefficient and the stored strain energy. Once this charge is determined, one can estimate the current and hence the power delivered to a load. MATERIALS AND STRUCTURAL SETUP Figure 1 shows the device under test. The crosssectional structure of the device is shown in Fig. 1a, Fig. 1b presents the interdigitated electrode topology, and Fig. 1c illustrates the mechanical stress and electric field distributions within the structure. The dimensions are defined as follow: g is the spacing between two adjacent fingers, w is the width of the finger, OL is the overlapping area, h is the piezoelectric thickness, and L is the length of the cantilever. The cantilevered beam/plate device is generally designed to resonate at a specific frequency of vibration of the structure on which the
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Table I. List of materials and their properties used in the simulations and model development Property
SiO2
ZnO
Pt
PZT
Elastic constant [GPa] Poisson ratio Density [10–15 kg/lm3] Dielectric constant Piezoelectric constant (d31 and d33) [pm/V]
70 0.17 2.15 5
80 0.3 5.68 9 5.4 and 11.3
145 0.35 2.14
85 0.4 7.5 80 23 and 60
device is mounted. The bending results in a strain distributed along the beam, which is then converted to electrical energy through the d33-mode piezoelectric effect. The electrode is formed in an interdigitated pattern on top of the piezoelectric thin film to generate strain parallel to the electric field, resulting in the d33 mode of the piezoelectric element. In this structure, SiO2 is used as the supporting layer, ZnO and Pb(Zr,Ti)O3 (PZT) are chosen as piezoelectric materials, and the IDT pattern is constructed from Pt. The thicknesses of the SiO2, piezoelectric material, and Pt are 0.2 lm, 0.2 lm, and 100 nm, respectively. The detailed material properties used in these simulations are listed in Table I. Although most simulations were done using ZnO as piezoelectric, subsets of simulations were also done using PZT for compare with published experimental data. SENSITIVITY AND PARAMETRIC MODELING METHODOLOGY Design of experiments is generally conducted to identify the number of finite-element simulations that is sufficient to provide objective statistical conclusions.15 The block diagram for the proposed algorithm is depicted in Fig. 2. The d33 cantilever analysis is split into two groups of independent parameters: the geometrical dimensions group, which contains three factors: electrode width, gap, and overlapping length; and the second group containing material parameters, namely the supporting layer, the piezoelectric layer (PZE), and metallization layer thicknesses. Two levels of each factor are chosen. The analysis ranges for all variables are presented in Table II, which shows three designs of experiments that have been conducted to fully study the effect of the mentioned parameters. Due to the large number of input variables, three DoEs are needed. CoventorWare was used to perform the multiphysics simulations.22 The Coventor simulations started with defining the structure parameters and their relative values. After collecting the data outputs (frequency and charge), they were then statistically processed by the analysis of variance (ANOVA) method to identify the most significant parameters. A factor, Fs, was used to compare the sum of the squares of the levels with the sum of the squares of the entire collected data.23 The Fs statistical factor can
Fig. 2. Block diagram of the proposed algorithm used in the analysis of the power output.
only assume nonnegative values. It is zero if all the sample means are identical and gets larger as they diverge. Once the statistically significant parameters were identified, the multiple linear regression (MLR) approach was applied to develop the model. To determine the relationship between the input variables (x) and the response variables, frequency (f), and charge output (Q), the following generalized multiple linear regression models are given by Ref. 24: ff ; Qg ¼
k X i¼1
b i xi þ
k X k X i¼1 j¼1
bij xi xj þ
k X
bjj x2j ;
(8)
j¼1
where the b0 are called regression coefficients and k is the number of variables.
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Table II. Summary of the factors, levels, and DoE and analysis of variance (ANOVA) analysis results for the simulations DoE Variable Level (2) Level (+) Units 1
g (X1) w (X2) OL (X3)
5 20 5 20 50 200 f SE ANOVA analysis for DoE-1
lm lm lm kHz nJ
2
hO (X1) hPZE (X2) hPt (X3)
0.2 1 0.2 1 20 100 f SE ANOVA analysis for DoE-2
lm lm lm kHz nJ
3
hO (X1) hPZE (X2) L (X3)
lm lm lm kHz nJ
0.2 1 0.2 1 250 750 f SE ANOVA analysis for DoE-3
Design of experiments treatments – 2.37 285
+ 2.35 115 Fs (Q)
– + – + – + + + + + + + + + 2.44 2.37 2.39 2.36 2.48 2.34 665 475 1175 400 2280 1230 X1 X2 X3 X1 9 X2 X1 9 X3 X2 9 X3 73.42 110.04 193.26 1.34 33.01 21.96 + + + + + + + + + + + + 1.49 4.6 3.75 6.42 1.37 4.34 3.7 6.29 9885 32 2600 22,385 2650 32 22,950 23,750 X1 X2 X3 X1 9 X2 X1 9 X3 X2 9 X3 Fs (f) 30,409 16,522 72 157 11 9 Fs (Q) 0.1 5.1 0.3 1.59 0.2 1.22 + + + + + + + + + + + + 4.64 16.5 13.87 24.5 9.14 11.48 9.7 17.1 15,000 2600 4000 3500 4200 3500 7100 6300 X1 X2 X3 X1 9 X2 X1 9 X3 X2 9 X3 Fs (f) 26 14 3.7 0.38 4.14 3.15 Fs (Q) 0.13 1.44 0.11 0.97 0.9 1.73
6
RESULTS AND ANALYSIS
5
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
4
Charge (pC)
Harmonic simulations were initially carried out to explore the deflection profile over frequency for a certain applied load. The Coventor-based Manhattan meshing model was chosen to mesh the d33 cantilever structure due to its orthogonal geometry, i.e., all model faces (patches) are planar and join at right angles (90°). The mesh sizes in the x, y, and z-directions were set to 0.5, 0.5, and 0.005, respectively. A sinusoidal load was applied to the tip of the top surface of the cantilever. The frequency of the load was swept from 100 Hz to 100 kHz. Having determined the fundamental mode of resonance, both the displacement and strain at the corresponding resonant frequency were recorded for different combinations. A series of DoEs were then conducted using ZnO as the piezoelectric to establish a direct relationship between the charge output and the geometrical dimensions and material parameters. The collected resonant frequency and strain energy for all DoEs are listed in Table II. The charge output corresponding to these strain energies was computed using Eq. 7 and is shown in Fig. 3. It was found that the amount of charge produced was proportional to the piezoelectric coefficient, the induced strain energy, the electrode gap, and the material thickness. The Coventor simulation results also showed that the resonant frequency was only mildly affected by changing the geometry of the IDT pattern. The variation in the resonant frequency among all the
3
2
1
0 10
100
1k
10k
Frequency (Hz) Fig. 3. Charge output as a function of frequency using Eq. (7) for different DoE-1 design combinations (Qn). The results show that the resonant frequency does not depend on the geometry, allowing one to optimize the electrode configuration for power output independently, without affecting the resonant frequency of the cantilever. The charge output strongly depends on the interdigitated electrode topology.
electrode design combinations was negligible. The fact that the electrode dimensions do not affect the resonant frequency adds a great deal of flexibility to
Modeling the Power Output of Piezoelectric Energy Harvesters
Q ¼ 8:2 103 ðwÞ 0:42 103 ðwOLÞ þ 0:37 103 ðgOLÞ þ 0:13 103 ðw2 Þ þ 0:5 103 ðg2 Þ; ½pC
ð9Þ
and for DoE-2, Q ¼ 1:3 102 ðhPZE Þ 0:108 102 ðhO hPZE Þ þ 1:21 102 ðhPZE hPt Þ þ 2:27 102 h2PZE ; ½pC (10) at resonant frequency of f ¼ 4:01ðhO Þ þ 3:08ðhPZE Þ 1:37ðhPZE hPt Þ½kHz; (11)
3000 2500 2000 1500 1000 500 0 0
5
10
15
20
25
30
35
40
Frequency step Fig. 4. Simulated charge output using PZT corresponding to DoE-3, over frequency steps using Eq. (7). The resonant frequencies corresponding to Q1 to Q8 are 4 kHz, 14 kHz, 10 kHz, 20 kHz, 0.46 kHz, 1.6 kHz, 1.2 kHz, and 2.3 kHz, respectively.
at resonant frequency of f ¼ 0:025ðLÞ 0:016ðhO LÞ 0:014ðhPZE LÞ þ 15 h2O þ 11:88 h2PZE ½kHz:
ð13Þ
hPZE, hO, and hPt, are the thickness of the piezoelectric, overlapping, and platinum layers. This equation set relates the charge output at a specific frequency to the material parameters, layer thicknesses, and dimensions of the structure. This analytical model can be used to optimize the IDT structure design and the cantilever performance without use of expensive software packages or lengthy simulation times. DoE-3 was carried out for the PZT materials. DoE-3 was performed with the most significant parameters including the length of the cantilever. The properties of the PZT materials are listed in Table I. The charge output simulations are shown in Fig. 4. The simulation and ANOVA analysis revealed that the charge output and resonant frequency exhibited the same dependency as in the case of ZnO. The charge output (QPZT ) and fPZT were computed using a multilinear regression model and were found to be, respectively, QPZT ¼ 2:87ðhPZE Þ þ 0:0036ðhPZE LÞ 1:5ðhO hPZE Þ ð14Þ 0:0013ðhO LÞ 2:2 h2PZE ½nC at resonant frequency of
whereas for DoE-3, Q ¼ 47:7ðhPZE Þ þ 0:01ðhPZE LÞ þ 2:86ðhO hPZE Þ 0:009ðhO LÞ 46:6 h2PZE ½pC
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Charge output (pC)
the design of the d33-based cantilever. This means that one can decouple the electrode geometry optimization from the other parameters. ANOVA statistical analysis was performed, and the results are presented in Table II. The results of the ANOVA analysis showed that the most significant parameters in terms of resonant frequency were the thicknesses of the supporting oxide layer, PZE layer, and cantilever length, whereas the charge output was more affected by the PZE thickness and the interdigitated electrode dimensions. The ANOVA analysis of DoE-1 presents the statistics for all three input variables of the IDT pattern and all two-factor interactions. Variables with higher values are more statistically significant. As shown in Table II, the highest value was obtained for the finger overlap (193.96), followed by finger width (110.04) and finger spacing (73.42). For DoE-2, Table II also shows that the supporting layer and PZE layer were the most significant for the resonant frequency, as evidenced by their large Fs values of 30,409 and 16,522, respectively. The metallization thickness had a very low statistic (72) and was therefore eliminated from further analysis, whereas in terms of strain energy, the PZE layer thickness was most significant, as shown by its large Fs value of 5.1. In the case of DoE-3, as shown in Table II, in terms of resonant frequency, the highest value was obtained for the supporting layer thickness (26), followed by the finger gap (14). For charge output, the PZE layer was the most significant, as evidenced by its large Fs value of 1.44. After the determination of the most significant parameters that influenced the resonant frequency and charge output, the multiple linear regression approach was applied to develop the model using Eq. (8), as follows: For the geometrical parameters in DoE-1, the charge output can be written as (see Fig. 1 for variable definitions)
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ð12Þ
fPZT ¼ 0:01ðLÞ 0:028ðhO LÞ 0:019ðhPZE LÞ þ 18:2 h2O þ 12:2 h2PZE ½kHz:
ð15Þ
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38
31
23
Current output
Charge (pC)
0.5
Model
15
Simulated
0.0
-0.5
8
-1.0
0 0
2k
4k
6k
8k
10k
Frequency (Hz)
0
10
20
30
40
50
Time step Fig. 6. Constructed normalized current in the time domain using Eq. (17) for the PZT cantilever.
Fig. 5. Comparison of results for synthesized energy strain obtained from the model Eq. (7) and from the Coventor simulation.
It is noticeable that the values of the estimated ~ depend on the material properties such coefficient b as density, Young’s modulus, and piezoelectric constants. The above-developed parametric models can be used to predict the charge output at any different parameter value combinations for either PZT or ZnO, provided that the DOE parameters are within the DOE factor limits. The MLR developed for the cantilever structure was used to validate the model accuracy by comparing the model results with those from the finite-element modeling (FEM). Figure 5 shows a comparison of the charge output obtained from the model Eq. (9) and from the Coventor simulations. Note that the accuracy of the analytical model is validated by the excellent match of the charge output obtained by our model with that simulated by Coventor. CURRENT AND AVERAGE POWER OUTPUT The model described above can be used to calculate the charge output from the d33 energy harvester. To estimate the power delivered to a load, the electric current output from the piezoelectric generator must be calculated. The induced strain energy is used to calculate the charge output in the frequency domain. The inverse Fourier transform is then used to compute the charge output time-domain signal. The inverse discrete Fourier transform (IDFT) is given by QðnÞ ¼ ð1=N Þ
N 1 X k¼0
2pi
QðkÞe N kn ;
(16)
where N is the number of the samples and Q(k) is the simulated frequency response. This signal could then be used to calculate the current output from the cantilever. The basic relationship in a capacitor is that the voltage is proportional to the charge on the plate. However, we need to know how current and charge are related. To derive this relationship, one needs to realize that the current flowing into the capacitor is the rate of charge flow into the capacitor, thus I ¼ dQ=dt:
(17)
The output current flowing in the capacitance of the harvester is shown in Fig. 6, which shows a transient response for a small period of time. The study of the transient dynamic characteristics of a PZT harvester utilizing electrical equivalent models has been performed in previous studies, and the model has shown fairly good accuracy under various conditions of mechanical stress.25,26 Figure 7 shows an electric equivalent circuit model for a PZT beam where a current source is connected in parallel with a resistor and a capacitor, forming a low-pass circuit. The capacitor represents the inherent capacitance of the device. After calculating the current flowing in the harvester, a load is now added in parallel to the harvester. The circuit can be described by using the current divider rule IR ¼
1 IT ; ð1 þ jxCRÞ
(18)
where IT is the generated current as represented in Fig. 8. x is the angular frequency, R is the load, and C is the harvester capacitance. Figure 8 shows the output current signals in the load for different RC
Modeling the Power Output of Piezoelectric Energy Harvesters
1483 0.35 200
Power (nWatt)
0.30
Power ( Watt)
0.25
150
100
50
0.20
0
0
0.15
20
40
Frequency step
0.10 Fig. 7. Circuit schematic showing the current divider rule used in this simulation.
0.05 0.00
0.50
0.4
=1.75 =1 =0.05
Normlizes signals
0.25
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Fig. 9. Power as a function of frequency for different RC time constant values. The inset shows the power delivered to the load versus frequency at the RC value indicated by the arrow.
0.00
Table III. Comparison of reported experimental power output of PZT piezoelectric energy harvesting devices versus the model developed in this study
-0.25
Ref. Choi et al.9 Jeon et al.10
-0.50
0
10
20
30
40
50
Time step Fig. 8. Current signal representing output from the cantilever for different RC time constants. s is the time constant.
values. The power delivered to the load versus the RC value based on this current computation is shown in Fig. 9. When the RC value is small, the maximum energy flows out of the PZT device. When the load resistance is increased, a large amount of energy is dissipated. However, when the load resistance becomes very high, the damping induced in the system decreases and therefore the ability to harvest energy from the PZT cantilever is reduced. The frequency spectrum of power generation around the resonant frequency of the generator is depicted in the inset to Fig. 9. A significant amount of unwanted damping will reduce the electrical power output. Table III shows a comparison of the calculated power output with previously reported experimental PZT thin-film cantilevers of comparable dimensions and thickness values. Compared with published thin-film results, the computed power is in the range that has been experimentally
Mode
Power
Load
d33 d33
1 lW 1 lW
5.2 MX 5.2 MX
demonstrated. This suggests that the simple model described above can be used to estimate the power output of piezoelectric cantilevers for any given geometry, material properties, and thickness, provided that these parameters fall in the range that was used to develop the analytical equations. CONCLUSIONS An algorithm based on design of experiments and finite-element simulations is presented. ANOVA revealed that the dependency of the charge output and natural frequency on the dimensions and materials of the piezoelectric cantilever are relatively decoupled. This decoupling adds significant flexibility in the design of the d33-based piezoelectric energy harvester. A circuit model was used to estimate the power delivered to a load. The computed power is in the range that has been experimentally reported. This suggests that the simple model described can be used to estimate the power output of piezoelectric cantilevers for any given geometry, material properties, and thickness, provided that these parameters fall in the range that was used to develop the analytical equations.
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