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Fundamental Understanding of Piezoelectric Strain Sensors

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Fundamental Understanding of Piezoelectric Strain Sensors Fundamental Understanding of Piezoelectric Strain Sensors JAYANT SIROHI* AND INDERJIT CHOPRA Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering, University of Maryland, College Park, MD 20742 ABSTRACT: This paper investigates the behavior of piezoelectric elements as strain sensors. Strain is measured in terms of the charge generated by the element as a result of the direct piezoelectric ef- fect. Strain measurements from piezoceramic (PZT) and piezofilm (PVDF) sensors are compared with strains from a conventional foil strain gage and the advantages of each type of sensor are dis- cussed, along with their limitations. The sensors are surface bonded to a beam and are calibrated over a frequency range of 5–500 Hz. Correction factors to account for transverse strain and shear lag ef- fects due to the bond layer are analytically derived and experimentally validated. The effect of tem- perature on the output of PZT strain sensors is investigated. Additionally, design of signal condition- ing electronics to collect the signals from the piezoelectric sensors is addressed. The superior performance of piezoelectric sensors compared to conventional strain gages in terms of sensitivity and signal to noise ratio is demonstrated. INTRODUCTION their superior noise immunity as compared to differentiated signals from conventional foil gages has been demonstrated. IEZOELECTRIC elements are commonly used in smart The correlation between the piezoelectric gage reading and Pstructural systems as both sensors and actuators (Chopra, the foil gage measurement is quite good; however the com- 1996). A key characteristic of these materials is the utiliza- parison was performed only at one frequency, 25 Hz. tion of the converse piezoelectric effect to actuate the struc- The work presented in this paper is an attempt to calibrate ture in addition to the direct effect to sense structural defor- piezoelectric strain sensors by comparing their calculated mation. Typically, piezoceramics are used as actuators and strain output to a conventional foil strain gage measurement. polymer piezo films are used as sensing materials. It is also Appropriate signal conditioning electronics is developed to possible to use piezoceramics for both sensing and actuation, collect the data from the strain sensor. The transfer functions as in the case of self-sensing actuators (Inman et al., 1992). In of both types of sensors are compared over a frequency range addition to the possibility of performing collocated control, from 5–500 Hz. such actuators/sensors have other advantages such as com- pactness, sensitivity over a large strain bandwidth and ease of embeddability for performing structural health monitoring PIEZOELECTRIC SENSORS as well as distributed active control functions concurrently. Many researchers have used piezoceramic sheet elements as Constitutive Relations sensors in controllable structural systems (Qui and Tani, 1995) and also in health monitoring applications (Samuel Under small field conditions, the constitutive relations for and Pines, 1997). Most of these applications rely on the rela- a piezoelectric material are (IEEE Standard, 1987): tive magnitudes of either the voltage or rate of change of volt- =+σ d σ age generated by the sensor, or the frequency spectrum of the DeEdijij im m (1) signal generated by the sensor. Several investigations have been carried out on discrete piezoelectic sensor systems (Qui εσ=+cE kjmdEjk s km (2) and Tani, 1995), active control of structures with feedback from piezoelectric sensors (Hanagud et al., 1992), and collo- which can be rewritten as cated sensors and actuators (Inman et al., 1992; Anderson and Hagood, 1994). However, a limited attempt has been È˘DEÈ˘edσ d È˘ made to accurately calibrate the magnitude of the measured Í˙= Í˙ Í˙ (3) Î˚εσÎ˚dscE Î˚ sensor voltage with actual structural strain. Piezoelectric strain rates sensors have been investigated by Lee and O’Sullivan (1991) and Lee et al. (1991) wherein where vector D of size (3 × 1) is the electric displacement (Coulomb/m2), ε is the strain vector (6 × 1) (dimensionless), σ *Author to whom correspondence should be addressed. E-mail: sirohij@ E is the applied electric field vector (3 × 1) (Volt/m)and m is eng.umd.edu the stress vector (6 × 1) (N/m2). The piezoelectric constants 246 JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 11—April 2000 1530-8138/00/04 0246-12 $10.00/0 DOI: 10.1106/8BFB-GC8P-XQ47-YCQ0 © 2001 Technomic Publishing Co., Inc. Fundamental Understanding of Piezoelectric Strain Sensors 247 σ are the dielectric permittivity eij of size (3 × 3) (Farad/m), the Equation (1) is the sensor equation and Equation (2) is the d c piezoelectric coefficients d im (3×6)andd jk (6×3) actuator equation. Actuator applications are based on the E (Coulomb/N or m/Volt), and the elastic compliance skm of converse piezoelectric effect. The actuator is bonded to a 2 c size (6 × 6) (m /N). The piezoelectric coefficient d jk (m/Volt) structure and an external electric field is applied to it, which d defines strain per unit field at constant stress and d im (Cou- results in an induced strain field. Sensor applications are lomb/N) defines electric displacement per unit stress at con- based on the direct effect. The sensor is exposed to a stress stant electric field. The superscripts c and d have been added field, and generates a charge in response, which is measured. to differentiate between the converse and direct piezoelectric In the case of a sensor, where the applied external electric effects, though in practice, these coefficients are numerically field is zero, Equation (3) becomes equal. The superscripts σ and E indicate that the quantity is σ measured at constant stress and constant electric field respec- È˘1 Í˙σ tively. For a sheet of piezoelectric material, the poling direc- È˘È ˘2 Dd1150000 0Í˙σ tion which is usually along the thickness, is denoted as the 3- Í˙Í= ˙Í˙3 Dd224000 00σ (8) axis and the 1-axis and 2-axis are in the plane of the sheet. Í˙Í ˙Í˙4 c Î˚ÎDddd3313233000 ˚Í˙σ The d jk matrix can then be expressed as 5 Í˙σ Î˚6 È˘00d31 Í˙00d This equation summarizes the principle of operation of pi- Í˙32 00d ezoelectric sensors. A stress field causes an electric displace- d = Í˙33 (4) Í˙00d24 ment to be generated [Equation (8)] as a result of the direct σ Í˙d15 00 piezoelectric effect. Note that shear stress in the 1-2 plane, 6 Î˚Í˙000 is not capable of generating any electric response. The electric displacement D is related to the generated charge by the relation where the coefficients d31, d32 and d33 relate the normal strain in the 1, 2 and 3 directions respectively to a field along the È˘ poling direction, E . The coefficients d and d relate the dA1 3 15 24 = Í˙ qDDDdAÚÚ []1232 (9) shear strain in the 1-3 plane to the field E1 and shear strain in Í˙ Î˚dA3 the 2-3 plane to the E2 field, respectively. Note that it is not possible to obtain shear in the 1-2 plane purely by application of an electric field. where dA1, dA2 and dA3 are the components of the electrode In general, the compliance matrix is of the form area in the 2-3, 1-3 and 1-2 planes respectively. It can be seen that the charge collected, q, depends only on the component of the infinitesimal electrode area dA normal to the displace- È˘SSS11 12 13 000 Í˙ment D. The charge q and the voltage generated across the SSS12 22 23 000 Í˙sensor electrodes Vc are related by the capacitance of the sen- E = Í˙SSS13 23 33 000 s (5) sor, Cp as Í˙000S44 00 Í˙0000S55 0 Í˙ VqC= / (10) Î˚00000S66 cp and the permittivity matrix is Therefore, by measuring the charge generated by the pi- ezoelectric material, from Equations (8) and (9), it is possible to calculate the stress in the material. From these values, È˘σ e11 00 knowing the compliance of the material, the strain in the ma- σσ= Í˙ e Í˙00e22 (6) terial is calculated. σ Î˚Í˙00e33 The sensors used in this work are all in the form of sheets (Figure 1), with its two faces coated with thin electrode lay- The stress vector is written as ers. The 1 and 2 axes of the piezoelectric material are in the plane of the sheet. In the case of a uniaxial stress field, the correlation between strain and charge developed is simple. È˘σσ È ˘ 111 However, for the case of a general plane stress distribution in Í˙σσ Í ˙ 222 the 1-2 plane, this correlation is complicated by the presence Í˙σσ Í ˙ σ ==Í˙333 Í ˙ of the d term in the dd matrix. σσ (7) 32 Í˙423 Í ˙ While any piezoelectric material can be used as a sensor, Í˙σσ Í ˙ 531 the present study is focused to two types of piezoelectric ma- Í˙σσ Í ˙ Î˚612 Î ˚ terials, and they are piezoceramics and polymer piezoelectric 248 JAYANT SIROHI AND INDERJIT CHOPRA Table 1. Typical properties at 25°C. PZT-5H PVDF Young’s modulus (GPa) 71 4–6 d31 (pC/N) –274 18–24 d32 (pC/N) –274 2.5–3 d33 (pC/N) 593 –33 e33 (nF/m) 30.1 0.106 Figure 1. Piezoelectric sheet. particular direction and finally poled. In the liquid phase, the individual polymer chains are free to take up any orientation film. The characteristics of each type of material are dis- and so a given volume of liquid has no net dipole moment. cussed below. After solidification, and stretching the film in one direction, the polymer chains are mostly aligned along the direction of PZT Sensors stretching.
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