Effective Electromechanical Coupling Research Articles

Radial vibration analysis for functionally graded ring piezoelectric transducers based on electromechanical equivalent circuit method

Ring piezoelectric transducers (rPETs) have attracted multitudinous attentions due to their widespread applications in hydro-acoustic engineering, structural health monitoring, and energy harvesting. However, cracks are inevitably occurred by the concentrated stress at the interface between metal and piezoelectric ceramics, which will seriously affect the performance and service life of the devices. Hence, a novel functionally graded ring piezoelectric transducer (FG-rPET) has been designed and an accurate analytical system is constructed mainly including a three-port electromechanical equivalent circuit model (EECM) which can acquire exact solutions in radial vibration. In addition, the proposed three-port EECM has excellent generalizability, which is also applicable to conventional rPETs. The validity of the EECM is verified by experiments and the finite element method. The vibration characteristics (resonance/anti-resonance frequencies and effective electromechanical coupling coefficient (keff)) with fundamental and second modes for the FG-rPETs can be regulated via the inhomogeneity index and wall thickness. Markedly enhanced keff at second modes can be obtained for the FG-rPETs with specific wall thicknesses. The proposed analytical system has important guiding significance for structure optimization design of piezoelectric devices, and the designed FG-rPET is expected to break the bottleneck of mechanical properties for the conventional rPETs.

Analysis on vibration characteristics of large-size rectangular piezoelectric composite plate based on quasi-periodic phononic crystal structure

Based on the theory of composite materials and phononic crystals (PCs), a large-size rectangular piezoelectric composite plate with the quasi-periodic PC structure composed of PZT-4 and epoxy is proposed in this paper. This PC structure can suppress the transverse vibration of the piezoelectric composite plate so that the thickness mode is purer and the thickness vibration amplitude is more uniform. Firstly, the vibration of the model is analyzed theoretically, the electromechanical equivalent circuit diagram of three-dimensional coupled vibration is established, and the resonance frequency equation is derived. The effects of the length, width, and thickness of the piezoelectric composite plate at the resonant frequency are obtained by the analytical method and the finite element method, the effective electromechanical coupling coefficient is also analyzed. The results show that the resonant frequency can be changed regularly and the electromechanical conversion can be improved by adjusting the size of the rectangular piezoelectric plate. The effect of the volume fraction of the scatterer on the resonant frequency in the thickness direction is studied by the finite element method. The band gap in X and Y directions of large-size rectangular piezoelectric plate with quasi-periodic PC structures are calculated. The results show that the theoretical results are in good agreement with the simulation results. When the resonance frequency is in the band gap, the decoupling phenomenon occurs, and then the vibration mode in the thickness direction is purer.

Lead-Free AE Sensor Based on BZT-BCT Ceramics.

In this study, an acoustic emission (AE) sensor was fabricated using lead-free Ba(Zr0.2Ti0.8)O3–0.5(Ba0.7Ca0.3)TiO3 (BZT–BCT) ceramics. The acoustic and electromechanical properties of the AE sensor were determined by the shapes of the piezoelectric ceramics. To optimize the AE sensor performance, the shapes of the ceramics were designed according to various diameter/thickness ratios (D/T) = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0. The BZT–BCT ceramic with D/T = 1.0 exhibited excellent values of a piezoelectric charge coefficient (d33), piezoelectric voltage coefficient (g33), and electromechanical coupling factor (kp), which were 370 (pC/N), 11.3 (10−3 Vm/N), and 0.58, respectively. Optimum values of resonant frequency (fr) = 172.724 (kHz), anti-resonant frequency (fa) = 196.067 (kHz), and effective electromechanical coupling factor (keff) = 0.473 were obtained for the manufactured BZT–BCT ceramic with D/T = 1.0. The maximum sensitivity and frequency of the AE sensor made of the BZT–BCT ceramic with a D/T ratio of 1.0 were 65 dB and 30 kHz, respectively.

Effect of acoustic load on electromechanical characteristics of sandwiched piezoelectric ultrasonic transducers in broken rail detection

This article investigates the effect of acoustic load on electromechanical characteristics of sandwiched piezoelectric ultrasonic transducers in time and frequency domains. The acoustic load includes aluminum and steel bars. It is shown that when the length of the acoustic load satisfies positive integer multiple of half-wavelength, the first resonant frequencies generate a smaller increase. With the increase of the length and characteristic impedance of acoustic load, the first resonant frequencies, and the tailing length of the received signal are increased, while the receiving signal amplitude, effective electromechanical coupling coefficient, and mechanical quality factor are decreased.

Influence of effective electrode coverage on the energy harvesting performance of piezoelectric cantilevers

Cantilevers are widely used for piezoelectric vibration energy harvesters (PEHs), thanks to the low resonance frequency and large mechanical strain under weak excitations. Electrode coverage optimization is essential for maximizing output power. In this study, bimorph cantilevers with segmented electrodes are developed to investigate the correlation between effective electrode coverage (EEC) and the harvesting behavior of piezoelectric cantilevers. A nonlinear electromechanical model is established using Hamilton's principle, numerically solved with the method of harmonic balance, and further validated by experimental measurements. Equivalent linear stiffness and equivalent mechanical damping ratio are firstly introduced for evaluating the equivalent effective electromechanical coupling coefficients and identifying strong and weak coupling harvesters. Both harvesters fabricated with and without proof mass behave mainly as strongly coupled systems and can obtain the maximum possible power at a very low EEC. Variation of EEC in higher range acquires almost constant maximum power on dramatically changing resistance. A comprehensive EEC design principle for acquiring the maximum output power is obtained for cantilevered PEHs. An optimal EEC obtaining the maximum effective electromechanical coupling coefficient is needed for acquiring the unique maximum power pick for weakly coupled harvesters, whereas a wide range of EEC can be used to maximize output power for harvesters those mainly behave as strongly coupled systems.

Fabrication and characterisation of piezoelectric thick-film microcantilever deposited on stainless steel using electrohydrodynamic jet deposition

This paper presents a lead zirconate titanate (PZT) thick-film microcantilever fabricated on a stainless-steel substrate using electrohydrodynamic jet (E-jet) deposition. PZT thick-film functional layer arrays were produced directly on stainless-steel substrates via E-jet deposition to form microcantilevers with a width of only 150 μm and thickness of 62 μm. The method can realise batch manufacturing of microscale cantilevers, improve the PZT active layer energy and microcantilever fracture toughness under vibrational deformation, and eliminate the need for complex photolithography and etching processes. The microcantilever accurately realised the correct mode shapes, consistent with simulation results. The vibrational amplitude of the microcantilever reached 5.14 μm, while Qm was up to 688.1, considerably larger than those of thick-film silicon-based microcantilevers and free-standing microcantilevers with similar sizes. An effective electromechanical coupling coefficient, Keff, of approximately 0.13 was obtained. The potential for efficient applications in microelectromechanical system piezoelectric devices was confirmed.

Ferroelectric Aluminum Scandium Nitride Thin Film Bulk Acoustic Resonators with Polarization‐Dependent Operating States

Given the recent discovery of ferroelectricity in highly doped AlN films and their widespread use in bulk acoustic wave (BAW) filters, the ferroelectric and acoustic performance of AlScN‐based devices at two distinct polarization states, i.e., nitrogen (N)‐polar and metal‐polar states, is investigated. Unreleased metal−ferroelectric−metal (MFM) capacitors, as well as suspended MFM structures in the form of thin‐film bulk acoustic resonators (FBARs), are fabricated based on Mo/AlScN/Mo‐sputtered films with ≈30% Sc concentration. The hysteresis polarization−electric field (P−E) loops and internal resistance of the MFM capacitors are characterized. The FBAR devices are characterized at the two polarization states, where the polarization inversion is accompanied by the modulation of thin‐film stress and resistance. Furthermore, the as‐fabricated N‐polar FBAR shows a fundamental thickness‐mode resonance frequency of 3.17 GHz, with a quality factor (QBode) of 572 and a high effective electromechanical coupling coefficient (kt2) of 11.4%. The FBAR frequency response changes when the polarization is switched from N‐polar to metal‐polar. Herein, the first investigations of electric and acoustic properties of N‐polar and metal‐polar AlScN‐based devices are demonstrated, paving the way for the realization of high‐kt2‐reconfigurable acoustic filters for next‐generation communication systems operating at the super‐high‐frequency (SHF) range.

Design and experimental study of longitudinal-torsional ultrasonic transducer with helical slots considering the stiffness variation

The ultrasonic transducer employing helical slots to generate longitudinal-torsional (L&T) ultrasonic vibration is popular in the field of hard and brittle material machining. However, the design approach for this type of L&T ultrasonic transducer is not well established. An improved design model combining equivalent circuit theory and material parameter equivalent method based on stiffness variation was proposed in this study. The structure parameters of the transducer can be obtained according to given resonance frequency and nodal position. The model can further predict electrical characteristics and mechanical responses under specific excitation. For the preliminary verification of this theoretical model and the optimization of the transducer, finite element analyses were carried out to investigate the influences of slot structure parameters on dynamic performances in terms of resonance frequency, frequency separation, and amplitude of L&T vibration. After that, a prototype was manufactured according to the design results, and the measurements of impedance curve and vibration amplitude were conducted. The experimental results indicate that this prototype resonates at 24,920 Hz with an effective electromechanical coupling coefficient of 0.19. Meanwhile, the vibration amplitude in longitudinal direction is 8.1 μm and that in tangential direction is 5.6 μm under 70-V sinusoidal voltage excitation, which are consistent with the theoretical design and simulation results. Furthermore, machining tests demonstrate that surface roughness reduction and surface quality improvement can be gained by adopting this L&T ultrasonic transducer, compared to using the longitudinal vibration transducer. The proposed design approach was validated and this study can provide guidelines for the design of L&T ultrasonic transducer for rotary ultrasonic machining (RUM).

Finite element modeling and analysis of adhesive layer effects in surface-bonded piezoelectric sensors and actuators including non-uniform thickness

A finite element model for multi-functional structures containing surface-bonded piezoelectric patches is proposed with two main objectives: to reduce the computational cost of detailed and accurate analyses; and to fully account for the adhesive layer between structure and patches, including non-uniform thickness. The proposed model is verified and used to assess the effect of adhesive layer properties on the performance of piezoelectric sensors and actuators, based on frequency responses and effective electromechanical coupling coefficient. Results indicate that increases of bonding layer thickness and thickness non-uniformity are not always detrimental to the actuation/sensing performance and could be explored as design parameters.

A Mathematical Model of a Piezoelectric Micro- Machined Hydrophone With Simulation and Experimental Validation

A mathematical model is necessary to analyze sensor performance and optimize its design. However, most of the existing models of piezoelectric micro-machined hydrophones are incomplete and too complex to understand. There has been no mathematical model for the sensitivity of piezoelectric hydrophones. To address this shortcoming, this paper proposes a mathematical model of a piezoelectric micro-machined hydrophone. The stresses in the piezoelectric layer and static displacement of the diaphragm under uniform sound pressure are derived. In addition, the resonant frequency and sensitivity of the hydrophone are analyzed. The finite element simulation is used to verify the validity of the proposed mathematical model. On the basis of the mathematical model and simulation, the diaphragm radius, top electrode area, and piezoelectric film thickness are optimized. Furthermore, the hydrophone is fabricated, and its morphology is observed. The impedance-frequency spectrum of the hydrophone is measured by HIOKI IM3536 LCR impedance analyzer. The tested resonant frequency is 0.5 MHz, which is relatively close to the calculated result of 0.537 MHz, so the validity of the proposed mathematical model is further validated. Also, the reason why the test result is less than the calculated result is explained. The effective electromechanical coupling coefficient derived from the impedance curve is 7.5%. The developed mathematical model can provide a useful guide for designing high performance piezoelectric micromachined hydrophone and make preliminary predictions of its characteristics.

Design and analysis of piezoelectric micromachined ultrasonic transducer using high coupling PMN-PT single crystal thin film for ultrasound imaging

Due to its higher frequency, improved impedance matching, and easy formation of array, piezoelectric micromachined ultrasonic transducer (PMUT) has drawn much attentions for application in ultrasound imaging. In this paper, PMUT based on [011] poled 0.70Pb(Mg1/3Nb2/3)O3–0.30PbTiO3 (PMN-0.30PT) single crystal thin film was proposed and modeled by performing finite element method. Influences of different configurations of top electrode shape and bottom cavity shape on PMUT performance were studied. The investigation results show that the resonant frequency of PMN-PT-based PMUT with circular top electrode and square bottom cavity is up to 27.155 MHz. The effective electromechanical coupling coefficient ( of PMUT was up to 2.32%, which was 127% larger than that of lead zirconate titanate-based PMUT. The static transmitting sensitivity was three times enhancement compared to PMUT with square top electrode and square bottom cavity. Furthermore, acoustic field characteristics of PMUT and PMUT arrays based on PMN-PT in water were investigated. The relative pulse-echo sensitivity level of 2 × 2 PMUT array was −35 dB with the reflector at 200 μm. The obtained results demonstrated that the proposed PMN-PT-based PMUT can achieve high frequency and large as well as considerable sensitivity, which is promising for application in high resolution ultrasound imaging.

Reusable piezo-sandwiched PTFE film for in-process monitoring of advanced composites

Environmental and cost-saving advantages derived from the use of composites attract aerospace and automotive industries as these materials offer significant structural and aerodynamic advantages over traditional metal structures. The composites industry, however, is concerned with the manufacturing processes as they cannot provide fast enough cycle time to match metal alloy processes. Our research aims to develop a sensing technology in the form of a reusable in situ cure monitoring and assessment system that can predict the formation of manufacturing defects and monitor the degree of cure. Thin-film material is chosen from various PTFE-based material by prioritizing the debonding effect and signal transmission through the composite part. Then, the film is used to sandwich piezoelectric actuators and sensors to monitor out-of-autoclave carbon fiber composite plates using ultrasonic Lamb waves by temporarily adhering to the manufactured part creating an effective electromechanical coupling between the sensing film and the laminate. Initial results, through the analysis of the fundamental antisymmetric A0 mode at low frequencies, indicate that analyzing the velocity and amplitude of these waves over cure time determines gelation and vitrification points. Experimental results have also proved the feasibility of using such a reusable film for different curing cycles, always determining certain cure parameters.

A dual-frequency piezoelectric micromachined ultrasound transducer array with low inter-element coupling effects

This paper presents a dual-frequency piezoelectric micromachined ultrasonic transducer (PMUT) line array with low crosstalk level, which was fabricated on silicon-on-insulator (SOI) wafers with sputtered piezoelectric thin film (PZT) and Si diaphragm structure. The obtained array consists of 120 of 0.77 MHz and 192 of 2.30 MHz PMUT units in total with minimum interspace of 50 μm. Due to the high piezoelectric coefficient of PZT, the PMUT shows high transmitting sensitivity in air and good effective electromechanical coupling factor. The displacement sensitivities are assessed to be 595 nm V−1 and 112 nm V−1 at the resonant frequencies of 0.77 MHz and 2.30 MHz respectively in air. To reduce the vibration coupling, rectangular grooves in the bottom silicon are designed between the adjacent line elements, and the PMUT units in the array are arranged in a mis-aligned style. Modal analysis for the 0.77 MHz units indicates neighbouring coupling-vibration decreases greatly from 44.5% to 14.8% of the excited vibration when the excited line is driven at 4Vpp, which proves both the grooves and the mis-aligned ranking are effective for coupling effect reduction. Moreover, results indicate the coupling effect between different frequencies can be ignored due to their inherent resonance characteristics. The sound pressures for a single 0.77 MHz line element and 2.30 MHz line element are evaluated to be 53 kPa and 73 kPa at a distance of 1 cm in water. This high performance dual-frequency PMUT line array makes some high resolution imaging methods possible based on PMUT technology, such as those of dual-frequency, total focus.

Modeling, fabrication, and structural characterization of thin film ZnO based film bulk acoustic resonator

This work reports the finite element modeling, fabrication and structural characterization a film bulk acoustic resonator (FBAR) devices using shadow mask technique. For the performance analysis of the FBAR device, simulations were performed on COMSOL Multiphysics platform. RF sputtering technique is used to deposit c-axis oriented zinc oxide (ZnO) thin film on Aluminum electrodes. The Aluminum electrodes were deposited using e-beam evaporation technique. The X-ray diffraction (XRD), atomic force microscopy (AFM), and scanning electron microscopy (SEM) techniques were used to examine the surface morphologies of the deposited piezoelectric film (ZnO). The quality of the deposited piezoelectric material film was measured with the help of XRD spectra and AFM is used to measure surface roughness of the deposited ZnO film and its measured value is 6.94 nm. The resonant characteristics of fabricated FBAR device are evaluated with the help of COMSOL Multiphysics simulations and the obtained values of effective electromechanical coupling constant (k2eff), quality factors (Qs and Qp) and Figure of merit are 3.0025%, 1811.3, 1711.4 and 54 (approximately), respectively.

A new broadband radial vibration ultrasonic transducer based on 2-2 piezoelectric composite material

Radial vibration transducer has the advantages of large radiation area, high radiation efficiency, uniform radial radiation, and wide range of action. Therefore, it is widely used in the technical fields of ultrasonic liquid treatment such as underwater acoustics, ultrasonic degradation and sonochemistry. On the other hand, the 2-2 piezoelectric composite material is one of the most commonly researched piezoelectric composite materials with the best development prospects. Compared with traditional pure piezoelectric ceramics, this new type of material has the advantages of low impedance, low mechanical quality factor, and frequency bandwidth. Therefore, in this paper we propose a new broadband radial vibration ultrasonic transducer based on 2-2 piezoelectric composite material, which is mainly composed of an inner metal ring and an outer piezoelectric ceramic composite ring. First, the Newnham series-parallel theory and the uniform field theory are used to derive the equivalent parameters of the 2-2 piezoelectric composite material. Second, the radial vibration of the combination of the metal ring and the radially polarized piezoelectric composite ceramic ring are analyzed by the analytical method. The six-terminal electromechanical equivalent circuit of the transducer is obtained, and the frequency equation of the transducer is also obtained. And then the relationship between the resonant frequency and anti-resonant frequency of the transducer, as well as the effective electromechanical coupling coefficient, geometric size, and two-phase volume ratio are analyzed. It is concluded that in order to obtain higher electromechanical conversion efficiency, the design of the transducer radius ratio should be as close as possible to 0.35. Although the higher proportion of polymer phase will lead the electromechanical conversion efficiency to decrease, it can also bring better acoustic matching ability. Therefore, the lower proportion of polymer phase can be selected in the transducer design. The finite element method is used to numerically simulate the radial vibration of the new transducer. The results show that the resonance frequency and anti-resonance frequency obtained by the analytical method are in good agreement with the numerical simulation results. In addition, the acoustic field of the transducer under water is simulated numerically. The results show that compared with the traditional pure ceramic radial transducer, the new composite radial transducer has a large emission voltage response amplitude, the working bandwidth is nearly doubled, and the acoustic matching is better.

Theoretical analysis and experimental validation of radial cascaded composite ultrasonic transducer* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11674206 and 11874253).

A radial cascaded composite ultrasonic transducer is analyzed. The transducer consists of three short metal tubes and two radially polarized piezoelectric ceramic short tubes arranged alternately along the radial direction. The short metal tubes and the piezoelectric ceramic short tubes are connected in parallel electrically and in series mechanically, which can multiply the input sound power and sound intensity. Based on the theory of plane stress, the electro-mechanical equivalent circuit of radial vibration of the transducer is derived firstly. The resonance/anti-resonance frequency equation and the expression of the effective electromechanical coupling coefficient are obtained. Excellent electromechanical characteristics are determined by changing the radial geometric dimensions. Two prototypes of the transducers are designed and manufactured to support the analytical theory. It is concluded that the theoretical resonance/anti-resonance frequencies are consistent with the numerical and experimental results. When R 2 is at certain values, both the anti-resonance frequency and effective electromechanical coupling coefficient corresponding to the second mode have maximal values. The radial cascaded composite ultrasonic transducer is expected to be used in the fields of ultrasonic water treatment and underwater acoustics.

Determination of effective properties of piezoelectric composites containing short fibers

The present paper studies the effects of local field fluctuations on the effective electromechanical coupling coefficients of piezoelectric composites containing short fibers. The finite-element method (FEM) is applied to study the overall behavior of such composites. Geometric models containing spheroid inclusions of two fiber arrangements, simple cubic and body-centered cubic, are modeled with FEM to predict the effective coefficients. The results thus obtained are validated with the results obtained from the analytical method based on Eshelby’s solution to the ellipsoidal inclusion problem. The analytical solution obtained through the Eshelby method is generalized for spheroid inclusions to compare the results with those of the FEM study. All finite-element studies are carried out through the finite-element software package Ansys 18.0. A detailed analysis of results is presented to understand better the behavior of such composites and their effective property variation with change in fiber volume fraction.

Modal effective electromechanical coupling coefficient of shear-mode piezoceramic sandwich cantilevers with segmented multicore: Experimental and numerical assessments

Smart sandwich cantilevers with aluminum faces and single and double cores, formed by assembled shear-mode piezoceramic patches with same poling, are experimentally and numerically assessed for the first time. To measure the electromechanical coupling efficiency of such vibrating smart structures, the so-called modal effective electromechanical coupling coefficient is used as a performance indicator. Hence, it is first experimentally analyzed under different electric connections (short circuit, open circuit, series wiring, and parallel wiring) of the patches’ electrodes; then, it is numerically investigated for models with different refinements (equipotential constraints and bonding adhesives) using ABAQUS® three-dimensional finite element simulations. It is found that the experimental modal effective electromechanical coupling coefficient is low for the smart shear-mode piezoceramic single core sandwich but can be increased using multilayer designs, as confirmed by the smart shear-mode piezoceramic double core sandwich. Numerically, it is found that the electric connection has less influence on the modal effective electromechanical coupling coefficient evaluation than the equipotential constraints and adhesives modeling, in particular for the smart shear double core sandwich. The proposed two benchmarks can be used by the research community of smart structures, systems, and devices for validating new shear-mode response-based theories and numerical models or designing related engineering applications, such as shunted damping, energy harvesting, structural health monitoring, resonators, and filters.

Large Coupling Acoustic Wave Resonators Based on LiNbO₃/SiO₂/Si Functional Substrate

In this work, wide band radio-frequency (RF) shear-horizontal surface acoustic wave (SH-SAW) resonators were designed and fabricated to attain a large effective electromechanical coupling (k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) over 35% near 1-GHz based on a thin-film lithium niobate (LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> /LN) on insulator layered substrate. In this study, the single-crystalline LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> thin film was bonded to a(100)-silicon carrier wafer with an intermediate silicon dioxide (SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) layer to form a simple and low-cost hetero acoustic impedance waveguide. Fabricated resonators with Au-electrodes show scalable wavelengths from 3.2μm to 4.4μm (770 to 1008 MHz) featuring k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> > 35% and Q of 250, which are sufficient for wide band RF filtering applications. Additionally, the potential of the SH-SAW resonator is demonstrated by a numerically synthesized ladder filter with a center frequency of 970 MHz, a 3-dB fractional bandwidth of 29.6%, and an insertion loss (IL) around 1.8 dB. It suggests the feasibility of developing wide bandwidth acoustic RF devices for potential 5G wireless communication through further design and fabrication optimizations.

Active area optimisation of film bulk acoustic resonator for improving performance parameters

In this Letter, an active area optimisation technique to improve the performance parameters of the film bulk acoustic resonator (FBAR) is proposed. The active area of the back trench membrane-based FBAR is optimised to remove the spurious modes, higher harmonic modes, and to confine the acoustic signal at its central part during the resonance. The effect of thickness variation of SiO2 layer on the performance parameters was studied using finite-element analysis (FEA) simulation. The SiO2 film, on silicon substrate, was used as the support layer and zinc oxide was used as the piezoelectric film for the resonator. The authors have successfully demonstrated the FBAR through FEA for an optimised active area of 320 × 320 µm2, series resonance frequency (f s) 1.249 GHz, and parallel resonance frequency (f p) 1.273 GHz with an effective electromechanical coupling coefficient ( k eff 2 ) of 4.65%.