Free Access
Issue
Eur. Phys. J. Appl. Phys.
Volume 87, Number 2, August 2019
Article Number 20401
Number of page(s) 11
Section Nanomaterials and Nanotechnologies
DOI https://doi.org/10.1051/epjap/2019190103
Published online 27 September 2019

© EDP Sciences, 2019

1 Introduction

In the last two decades, resonating microstructures have been increasingly used in sensors such as chemical and biochemical sensors [14], ultrasensitive mass sensors [58], pressure sensors [9] and fluid density and viscosity sensors [10]. Piezoelectric MCs are among the microstructures utilized as sensors [1114]. The piezoelectric layer in such MCs operates as both actuator and sensor, thus reducing the size and response time. Such sensors are used with one or two piezoelectric layers. In a unimorph MC, the piezoelectric layer acts as actuator or sensor. Two piezoelectric layers facilitate the simultaneous use of a MC as both actuator and sensor [1517]. The use of two piezoelectric layers not only miniaturizes the system, but also makes it a more sensitive sensor by increasing the MC stiffness. In such MCs, one of the piezoelectric layers works as a vibration actuator (i.e. an actuator layer). The output current of the piezoelectric layer is used as the measurement parameter, which is why the piezoelectric layer is called as the sensor layer. The output current is generated via the Wheatstone bridge. The main purpose of designing this type of sensor is maximizing the piezoelectric output current. Neethu and Suja [18] designed a MC biosensor to measure the mass absorbed by the MC, based on a piezoresistive mechanism. They studied the effect of geometric dimensions and MC materials on piezoresistivity. In spite of the great attention to optical sensors in the last two decades, recent self-sensing systems implemented in MEMS systems offer higher accuracy and performance. Bausells [19] developed a self-sensing MC which was as accurate as optical sensor systems, thus achieving both sufficient sensing accuracy and miniaturization. Rakotondrabe et al. [20] proposed a self-actuating, self-sensing system based on piezoelectric beams which is able to measure both deflection and force signals. Faegh et al. [21] used a self-sensing piezoelectric MC as a mass sensor. They modeled MC vibration in higher vibrating modes to study the influence of nanoparticle absorption on the resonance frequency shift. Ruppet and Mohemani [22] used a piezoelectric MC as self-actuating and self-sensing system in multimode actuation. The results of dynamic modeling of the MC indicated more than two orders of magnitude increase in deflection to strain sensitivity on the fifth eigen mode leading to a remarkable signal-to-noise ratio. By using the electrode segmentation, Hoffman and Twiefel [23] acquired more information about flux distribution in the piezoelectric material, and in turn could measure the mode shape of MC. Zhao et al. [24] implemented a self-sensing MC in liquid in addition to electromagnetic actuation to measure liquid density. They investigated the influence of MC geometry and operating mode on sensor sensitivity.

Chemical, biochemical, and mass sensors are usually implemented in liquid environments. Since piezoelectric MCs offer proper resistance to environmental variations and do not cause turbulence in the liquid environment, they are a good candidate for such applications. Studying the influence of a fluid's hydrodynamic force on the measurement parameter (piezoelectric layer output current) in such sensors reveals the effect of liquid environment on this parameter in addition to helping with better designing the sensors. Due to the difficulty of accurately calculating hydrodynamic force, a number of models have been proposed for dynamic modeling of MCs in liquid environments to simplify hydrodynamic force equations [2528]. These models have been evaluated and implemented in many studies [2931].

Considering the extensive applications of self-sensing sensors, this paper investigates the behavior of a piezoelectric MC in self-actuating and self-sensing modes. As the piezoelectric layer output current in self-sensing mode is used as the measurement parameter, the design of such sensors is highly influenced by this parameter, that is, an optimal sensor requires maximizing the piezoelectric layer output current with each actuation. Due to the extensive application of such sensors in liquid environments, this study presents a dynamic analysis of MCs in liquid environment. The Euler-Bernoulli theory and sphere-string model are utilized to obtain the differential equation for MC vibration in liquid environment. Then the differential equation for the MC vibration and the piezoelectric equations are utilized to obtain the Wheatstone bridge current. After that, the influence of the geometric dimensions of MC on the output current is analyzed via Sobol sensitivity analysis method.

2 Dynamic modeling of piezoelectric MCs motions

In piezoelectric MCs used in micro-electromechanical systems, the deformations and the cross-section rotations are significantly smaller than the length of the MC. Similarly, the cross-sectional area is negligible compared to the length of the MC. Therefore, the geometric and inertial nonlinearities as well as the rotational shear and inertial deformations are negligible, therefore the Euler–Bernoulli beam model can be used for analysis.

For dynamic modeling of the MC, two piezoelectric layers of equal lengths were implemented on both sides of the MC (Fig. 1). Each piezoelectric layer was enclosed by two electrodes. The two electrodes on the sides of the actuating piezoelectric layer were connected to an AC voltage source power. The other two electrodes were connected to a Wheatstone Bridge to transform the potential difference on the two sides of the piezoelectric sensor to electric current. The potential strain energy of the MC can be expressed as:(1)

In equation (1), Cv  = Ce P(t), where P(t) is the input voltage to the piezoelectric actuator layer. K 1 and K 2 are the stiffness in each part of the MC, and Ce is the electromechanical coupling coefficient. They can be written as:(2) (3) (4)

In these equations, i and j subscripts are used to specify MC layers. The layers are numbered from the lowest to the highest. E, h, and w denote Young's modulus, thickness, and width of each layer, respectively. The kinetic energy of the MC in a flexural vibration is expressed as:(5)where m 1 and m 2 represent the mass per unit length of the MC in each part, respectively. According to Figure 1, these parameters are obtained as:(6) (7)where ρi is the density of each layer. Assuming the force exerted on the unit length of the MC is denoted by f (x, t), the virtual work caused by the non-conservative force (Wnc ) can be expressed as:(8)

As the MC vibrates in a liquid environment, f (x, t) is defined as the distributed hydrodynamic fluid force exerted on the microcantilever surface. Given the difficulty of determining this force, numerous studies have attempted to model these hydrodynamic forces applied to beams [29,31]. These efforts mainly aimed to present a simplified and practical model to describe the hydrodynamic force exerted on the beams. Among the proposed models, the model of string of spheres offers a high precision. This model is more efficiency in modeling the hydrodynamic forces exerted on MCs with geometric discontinuities [30]. In this method, in order to model the hydrodynamic fluid force, the MC is considered as a set of attached spheres. The force exerted on these spheres during their movement in the liquid environment is considered as the hydrodynamic force exerted on the MC. The hydrodynamic force applied on the sphere floating in the viscous fluid can be expressed as [26]:(9)

where v is the displacement of the sphere modeled in the fluid, η is the viscosity, ρliq is the fluid density, ω vibrating frequency and R is the sphere radius. Moreover, δ is the penetration depth of the acoustic wave obtained as:(10)

Based on the string of spheres model, the hydrodynamic force exerted on the unit length of the MC can be considered equal to the sum of the forces applied on the spheres in a unit length of the MC. Thus, the hydrodynamic force applied to the MC can be expressed as [26]:(11)

Based on equation (11), the added mass and damping due to the presence of the fluid can be written as:(12) (13)where cliq is the hydrodynamic damping coefficient. According to the potential strain energy, kinetic energy and virtual work equations as well as the Lagrange equation, the governing equation for the MC motion in liquid environment is derived as:(14)where p(t) is the input voltage of the actuating piezoelectric layer, and m(x), K(x), and Ce (x) are:(15) (16) (17) (18)where H(x) is Heaviside function and Ci is the intrinsic damping of the MC. Using the Galerkin linear approximation, the flexural deformation of MC versus time and space can be decoupled as:(19)where ψi (x) denotes the nth MC mode shape and qi (t) is the generalized time-dependent coordinate. Considering the effect of the geometric discontinuity of the MC on the modes, ψi (20)

where , ,, , and  are unknown coefficients that can be obtained by applying the boundary conditions of MC, continuity conditions and mass normalization. According to Figure 2, the MC is fixed at one end and free at the other. The boundary conditions, therefore, are:(21)

Given that the MC maintains its continuity under vibration, the continuity conditions for deformation, slope, bending moment, and shear force at x = L 1 (the end of the piezoelectric layer) can be stated as follows:(22)

The unknown coefficients can be obtained by solving the characteristics equation and using a normalization condition with respect to mass as follows:(23) (24)where δnm is Kronecker function. In order to obtain the natural frequency of the MC, the determinant of the unknown coefficients in equations (22) and (24) (characteristic equation) can be used: (25)

where(26)

By substituting equation (14) into equation (10), multiplying the two sides of the equation with ψ i (x), and integrating along the beam length, the following ordinary differential equation is obtained:(27)where(28) (29) (30)

Numerical methods can be employed to solve equation (27). To this end, the Runge-Kutta method was used and programmed in MATLAB.

In the MC with two piezoelectric layers, one layer is used as the actuator and the other as a sensor. The MC vibrates as an AC voltage is applied to the piezoelectric layer. With the transverse flexure of the piezoelectric layer during vibration, an electric voltage is induced in this layer. This voltage can then be converted to electric current by connecting the Wheatstone bridge to the electrodes of this layer. Electric displacement in the z direction in the piezoelectric layer is expressed as a function of strain along the x direction and electric field in the piezoelectric layer:(31)where ezx is the piezoelectric constant along the coupled direction zx, zz is the dielectric constant, and Ez is the electric field along z direction in the piezoelectric layer. The electric charge generated on the electrode surface can be expressed by integrating the electric displacement in this area:(32)Since the both sides of the piezoelectric layer are confined by two electrode layers, the electric potential of the surface is the same as the potential difference generated in the electrodes. Assuming a constant electric field and denoting the potential difference between the upper and lower surfaces of the piezoelectric layer by P, the electric field can be expressed as:(33)

By substituting equation (33) into equation (32), one can write:(34)where  is the beam slope and φ (L 1) is the end slope of the piezoelectric layer, dzx is piezoelectric constant, Zp is the distance from the plane of zero strain to the neutral plane of the piezoelectric layer. Since the differential of the charge induced on the electrode surface is equal to the ratio of the output current to the external impedance, the amplitude of the electric current generated from the bending of the piezoelectric layer is obtained from multiplying the charge with the frequency:(35)where ω is MC frequency. In electric circuits, current is resulted from division of the voltage by the electric resistance. Therefore, according to equations (34) and (35), the amplitude of the current can be obtained as:(36)

thumbnail Fig. 1

Piezoelectric MC schematic.

thumbnail Fig. 2

Effect of external electrical resistance on generated electrical power.

3 Simulation and discussion

To simulate the analysis, the MC was considered with two piezoelectric layers (called the actuator and sensor layers) with the same length, which was shorter than the base layer. By using the Wheatstone bridge in the output of the piezoelectric electrodes of the sensor, the output electrical current and the power of this layer can be measured. The mechanical characteristics and geometrical features of the MC layers are given in Table 1.

Table 2 shows the output current of the sensor layer for the input voltages 25 mV and 12.5 mV, comparing the analytical results with the experimental results obtained by Itoh et al. [32]. The results indicate an acceptable accuracy for the analytic method used to calculate the output current of the piezoelectric layer.

Such energy consumers as the Wheatstone bridge are needed in piezoelectric sensors to convert voltage produced into an electrical current, thereby generating electrical power. The electrical power generated by the piezoelectric layer not only depends on the electrical resistance of the Wheatstone bridge, but also on the piezoelectric layer material. Figure 2 shows the effect of the Wheatstone bridge resistance and piezoelectric layer material on the output power of this layer. According to this figure, the output power of the piezoelectric layer is greatly affected by external resistance. To achieve a maximum output power, it is very important to have a resistance proportional to the piezoelectric layer material. The maximum output power obtains for the PZT-4, PZT-5A, and PZT-5H layers in the resistance of 18, 14, and 7 kΩ, respectively. The simulation results also show that the output power of the piezoelectric layer increased with increasing the piezoelectric constant. In that, the PZT-5H generated the highest electrical power because of its greatest piezoelectric constant.

Since the micro-sensors are used in different environments, investigation and comparison of their performance in different environments are very important. To investigate the effect of environment on generated electrical power, air, water, and water-glycerol solution are considered. According to the results of Figure 3, achieving higher electrical power is more plausible in air than in a liquid environment. The voltage induced to both sides of the piezoelectric layer reduces with increasing the density and viscosity of the fluid, in which the MC is vibrating, due to the reduction of vibrating motion and MC slope at each point. Therefore, the generated electrical power reduces with increasing the liquid density. Figure 3 also shows that the optimal resistance increases with increasing the liquid density, in that the optimal resistance is 6, 14, and 15 kΩ in the air, water, and water-glycerol solution.

Table 1

Geometrical dimensions and mechanical properties of simulated MC.

Table 2

The effect of the input voltage on the output current of the sensing piezoelectric layer.

thumbnail Fig. 3

Effect of environment on the generated electrical power.

3.1 Sensitivity analysis of output electrical current

Since the output electrical current of piezoelectric layer is used as measuring parameter in self-sensors, the higher the output current of the piezoelectric layer, the better the sensor accuracy and the lower its noise. The geometric dimensions of the MC are significantly affect parameters on the output electrical current of the piezoelectric layer. As a result, the maximum output electrical current of the piezoelectric layer and subsequently the best sensor performance can be achieved by selecting appropriate geometric dimensions during the sensor design and fabrication processes. The Sobol's sensitivity analysis can be used to investigate the effect of geometric dimensions of a self-sensor on the output electrical current. This method allows for investigating the effect of a number of parameters on the main parameter.

Figure 4 shows the effect of each MC layer's thickness on the output electrical current in the first three vibrating modes. According to the results (Fig. 4a), it can be concluded that the output electrical current of piezoelectric layer in the first to third modes is maximized, respectively, at the thickness of 0.52, 0.50, and 0.56 μm. These dimensions are the best thickness for designing the top electrode of this multi-layer MC. Figure 4b shows that the output electrical current in the first and second vibrating modes reduces with increasing the thickness of piezoelectric layer used as the sensor. In the third vibrating mode, this current is minimized at the thickness of 2.5 μm. In other words, a lower value of the layer thickness in the first two vibrating modes is more appropriate in designing this kind of sensor. According to Figure 4c, the output electrical current of the piezoelectric layer reduces with increasing the electrode thickness between the sensor layer and main layer of the MC. In that, the output electrical current is minimized in the first mode and the thickness of 0.72 μm, which is the worst value of thickness. Figure 4d shows an increased output electric current of the base layer in the first and third modes. In addition, the electrical current is maximized in the second mode in the thickness of 3.9 μm. Figure 4e shows the effect of the thickness of the piezoelectric layer used as actuator on the output current of the sensor piezoelectric layer. According to this figure, the output electrical current of the piezoelectric layer in the first to third vibrating modes increased with increasing the thickness of the piezoelectric actuator layer.

Figure 5 shows the effect of MC layer length on the output electrical current. The output electrical current in the first and second vibrating modes reduces with increasing the MC length, and is minimized in the third mode and the length of 260 μm (Fig. 5a). Moreover, the output electrical current increases in the first and second vibrating modes with increasing the piezoelectric layer length. The output current is maximized in the third mode and the thickness of 120 μm (Fig. 5b).

The MC width is one of the important parameters inhibiting MC vibrating motion in liquid environment. As shown in equation (11), the hydrodynamic force applied to the MC is directly related to its width, therefore increasing the width also increases the hydrodynamic force. As this force opposes the vibration motion, its increased value reduces vibrating amplitude and the MC slope at any point. According to equation (36), reduction in slope of MC reduces the output electric current. Figure 6a shows the output electrical current reduction with increasing the width in the first three vibrating modes. This is somehow different with respect to the piezoelectric layer width. In piezoelectric MCs, the piezoelectric layer width cannot exceed the MC width. As a result, the width of this layer has no marked effect on vibrating motion damping. The piezoelectric layer width not only increases the actuating ability and output electrical current of this layer, but also reduces deformation and output electrical current with increasing the MC stiffness. In this way, it is relatively difficult to predict the effect of this layer's width on the output electrical current. Figure 6b shows that the output electrical current of the sensor layer is minimized in the first to third modes and the widths of 20, 18, and 22 μm, respectively. However, Figure 6c shows that the output electrical current in the first mode increases with increasing the actuator layer. In addition, this current is maximized in the second and third modes and the widths of 24 and 23 μm.

Figures 46 show that in the first three vibrating modes of MC, for the majority of dimensions, second mode has the highest output current. Therefore, it is the best mode to achieve maximum output electrical current from the piezoelectric layer.

As the geometric parameters of the MC greatly influence the output current of the piezoelectric layer, these parameters have significant importance in designing and manufacturing piezoelectric MCs. Undoubtedly, the geometric parameters which have a greater impact on the output electrical current can play as the key parameter in designing such MCs. Figure 7 presents the sensitivity of the output electrical current to geometric parameters of MC layers. According to Figure 7a and b, the geometric dimensions with the greatest impacts on the output electrical current in the first and second modes include piezoelectric layer length, followed by the base layer length, and piezoelectric layer thickness. Moreover, the geometric dimensions with the greatest impacts on output electrical current in the third mode are MC length, followed by the piezoelectric layer length and actuator layer thickness (Fig. 7c).

thumbnail Fig. 4

Effect of thickness of MC layer on output electrical current; (a) sensor layer, (b) sensor's bottom electrode, (c) base layer, (d) sensor's top electrode, (e) actuator layer.

thumbnail Fig. 5

Effect of length of layers on output current; (a) base, (b) piezoelectric.

thumbnail Fig. 6

Effect of width of piezoelectric layers on output electrical current; (a) base layer, (b) sensor layer, (c) actuator layer.

thumbnail Fig. 7

Sensitivity of output electrical current to geometric dimensions, (a) first mode, (b) second mode, (c) third mode.

4 Conclusion

Piezoelectric MC sensors are a new, high-performance generation of sensors used in mechanical micro/nano-systems. Studying their sensing behavior can be of assistance in improving their performance and functionality. This paper studies the dynamics equation for the piezoelectric MC vibrations in self-sensing mode. In a self-sensing piezoelectric MC, the output current of the piezoelectric layer is used as the detection parameter. Indeed, the higher the output current of the piezoelectric layer in a specific actuation, the lower the system noise and energy consumption for achieving a higher electric current. Given the impact of MC dimensions on the output current of the piezoelectric layer, the correct selection of dimensions is of great significance in the optimal design of this type of sensor. The simulation and sensitivity analysis of the vibrations of a multilayer piezoelectric MC indicated the following results:

  • the output power of the piezoelectric layer was influenced by the electrical resistance of the Wheatstone bridge. The maximum electrical power occurred at different electrical resistances based on the type of the piezoelectric material used in MC. For example, the maximum electrical power occurred at resistances of 18, 14, and 7 KΩ for PZT-4, PZT-5a, and PZT-5H, respectively. The specified resistances can be applied to the selected Wheatstone bridge resistance in this type of sensor since the aim in the piezoelectric sensor is to achieve maximum electric power;

  • the performance of piezoelectric MC is better in the second vibrating mode (between the first three vibrating modes) as it generated more electric current;

  • the effect of hydrodynamic force on the MC vibration is reduced by reducing the fluid density and viscosity. This way, the MC can vibrate with larger amplitude, thus enhancing electric current generation at the piezoelectric layer. Accordingly, in cases where sensing is necessarily done in a liquid environment, using a liquid with lower density helps improve the performance of the sensor.

  • among the geometric parameters of MC, the MC length and piezoelectric layer length are the most influential parameters on the output current in the first three vibrating modes which makes them of significance in the design process.

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Cite this article as: Zahra Nadimi Shahraki, Reza Ghaderi, Vibration and sensitivity analysis of piezoelectric microcantilever as a self-sensing sensor, Eur. Phys. J. Appl. Phys. 87, 20401 (2019)

All Tables

Table 1

Geometrical dimensions and mechanical properties of simulated MC.

Table 2

The effect of the input voltage on the output current of the sensing piezoelectric layer.

All Figures

thumbnail Fig. 1

Piezoelectric MC schematic.

In the text
thumbnail Fig. 2

Effect of external electrical resistance on generated electrical power.

In the text
thumbnail Fig. 3

Effect of environment on the generated electrical power.

In the text
thumbnail Fig. 4

Effect of thickness of MC layer on output electrical current; (a) sensor layer, (b) sensor's bottom electrode, (c) base layer, (d) sensor's top electrode, (e) actuator layer.

In the text
thumbnail Fig. 5

Effect of length of layers on output current; (a) base, (b) piezoelectric.

In the text
thumbnail Fig. 6

Effect of width of piezoelectric layers on output electrical current; (a) base layer, (b) sensor layer, (c) actuator layer.

In the text
thumbnail Fig. 7

Sensitivity of output electrical current to geometric dimensions, (a) first mode, (b) second mode, (c) third mode.

In the text

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