DESIGN, MODELING, AND CONSTRUCTION OF A LOW FREQUENCY BIMORPH-PIEZOELECTRIC ACCELEROMETER F. García, C. I. Huerta,1 H. Orozco,2 and E. L. Hixson3 Exploración y Producción/Geofísica de Explotación-IMP, Eje Central Lázaro Cárdenas # 152, 07730 México, D.F. Tel/Fax. (525) 567-5476. Email:
[email protected] RESUMEN En este artículo, el modelo analítico y experimental de un nuevo tipo de acelerómetro piezoeléctrico a bajas frequencias es desarrollado. El diseño del acelerómetro está basado en el efecto piezoeléctrico y en las leyes de transductores o convertidores de energía electromecánica, representados con teoría de dos y tres puertos. La función de transferencia es obtenida analíticamente y con la ayuda de un simulador de circuitos eléctricos. A fin de caracterizar y calibrar nuestro prototipo, un analizar de espectros dinámico y un acelerómetro estándar fue utilizado. Posibles aplicaciones industriales para este tipo de acelerómetro son discutidas. ABSTRACT In this paper, the analytical model for a new low frequency Bimorph-piezoelectric accelerometer is developed and validated experimentally. The design is based on the piezoelectric effect and in the transducer laws, represented by two and three-port electromechanical equations. The desired relationship between voltage and free acceleration is obtained by using electrical and mechanical circuit analogies, and an electrical circuit simulator. A dynamic signal analyzer and a reference accelerometer were used to calibrate our prototype. Possible industrial applications are pointed out. 1. INTRODUCTION There are many types of sensing elements to detect acceleration. The most common ones are capacitive, piezoresistive, piezoelectric, thermal, surface acoustic wave, and electromagnetic [1]. We decided to use the piezoelectric one because it offers advantage for measuring vibratory acceleration occurring over a wide range of frequencies and amplitude. In addition, since the piezoelectric effect generates free charges, piezoelectric accelerometers do not need bias voltage and have good off-axis noise rejection, high linearity, and a wide temperature range (up to 120 °C). An accelerometer can then be defined as an electromechanical transducer that has an output voltage proportional to acceleration. The accelerometers always operate below their resonance frequency. Thus the resonance frequency must be high. The useful “flat” range, where the most accurate measurements can be made is shown in Figure 1, which is a typical frequency response of an accelerometer. The flat range in this case is from 0.01 to 150 Hz at ± 3dB. Within the flat region the calibration factor is a constant.
Figure 1. Frequency response and useful flat range for an accelerometer. 1
Departamento de Sismología, CICESE, Ensenada, B.C. México. Email:
[email protected] Departamento de Ingeniería Mecánica, Instituto Tecnológico de Celaya, Celaya, Gto, México. Email:
[email protected] 3 Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, 78712 USA. Email:
[email protected] 2
2. THEORY AND BASIC CONCEPTS Piezoelectricity is “pressure electricity”. When an external force strains the piezoelectric elements, displaced electrical charge accumulates on opposing surfaces thus it becomes a force sensor. In addition to natural quartz crystals, artificially man-made ceramics are used. These materials become piezoelectric by the application of a large polarizing electric field. Some common materials used as electromechanical transducers include Barium Titanate and variations of Lead Zirconate and Lead Titanate (PZT). “Bimorph” is the name for a flexible-type piezoelectric element, which has the capacity for handling larger motions and smaller forces than single piezoelectric plates. The Bimorph consist of two transverse-expander plates of ceramic, secured together face to face in such manner that a voltage applied to the electrode causes the plates to deform in opposite directions, resulting in a bending action. Conversely, mechanical bending of the element will cause it to develop a corresponding voltage between electrode terminals. Many different sizes and shapes of piezoelectric force sensors can be used to sense acceleration. By measuring the force to a seismic mass acceleration is sensed by virtue of Newton’s 2nd law: A = F/M. The different element configurations are shown in Figure 2. The compression design features high rigidity, making it useful for implementation in high frequency applications. Its narrow frequency range offsets the simplicity of the flexural design. The shear configuration offers a well-balanced blend of wide frequency range, low off axis sensitivity, and low sensitivity to base strain. M as s
P i e z o e l e c t r ic c r y s t a ls M as s
C o m p r e s s io n
F le x u r a l
Shear
Figure 2. Material Configurations. 3. ANALYTICAL MODELING The design used here to generate a signal proportional to acceleration is shown by Figure 3. A bender Bimorph element was mounted as a cantilever beam with an external mass (M1) at the free end or port 1, and it is clamped at the other end (port 2). Bender Bimorph M1 Motion Test structure or shaker
Signal out
External mass Port 1 free with mass M1
Port 2 clamped with aluminum pads (electrodes)
Figure 3. Bimorph accelerometer for generating a signal proportional to acceleration. According to the Fischer [2] and Hixson [3] approach, the piezoelectric accelerometers are transducers based on the following principles of conversion of energy Clamped force
Fc = α me E
Short circuit current I sc = α emV These lead naturally to the following general two-port equations
(1) (2 )
v I = Ye E + α emV
(3)
e F = α me E + Z mV
(4 )
The αme factor, is known as the electromechanical factor of the ideal transformer used to represent the coupling of the mechanical port with the electrical one. Since this type of transducers are reciprocal, αem = αme, we can then expressed it just as α. Eqs. (3) and (4) include an electrical and mechanical circuit. In the complete analysis we must also include the amplifier or instrument input resistance (RL) and capacitance (CL), as well as that for the cable. The mechanical representation for the piezoelectric accelerometer is shown by Figure 4. Port 1
Port 2, clamped end with mass M2
Port 1, free end with mass M1
Port 2
Acceleration (Ao) reference
F2, V2
F1, V1
M1’
Signal out (E0)
Series type Bimorph
Where
Cm
e
M2’
M1’ = M1 + Mc/2 = 2.1 gr. M2’ = M2 + Mc/2 = 57.1 gr.
reference
Figure 4. Accelerometer’s mechanical circuit. Where Cme is the Bimorph mechanical compliance, M1’ and M2’ is the Bimorph effective mass plus the external mass M1 attached on it. The electrical parameter is Cev. Rm is the mechanical resistance due to internal friction of the Bimorph. As we can see from Figure 4, what we really have is a 3-port device, two mechanical ports and one electrical port. The complete electromechanical model for the piezoelectric accelerometer is described by the following energy conversion 3-port equations v (5) I = Ye E + αV1 + αV2
(6) (7 )
F1 = αE + Z11V1 + Z12V2 F2 = αE + Z 21V1 + Z 22V2
V1
V2 α(V1 + V2)
I
1:α
(V1 + V2)
F1 Eo
F2
v
Ye
F = αEo
Figure 5. 3-Port electromechanical model for piezoelectric accelerometers. The electrical impedance or admittance must be measured for both mechanical ports clamped. The mechanical parameters are represented by Z-parameters measured with the electrical port shorted. Figure 5 shows the 3-port representation for the piezoelectric accelerometer that satisfies Eqs. (5) – (7). The impedance parameters are Za = Z11 – Z12, Zb = Z22 – Z21, and Zc = Z12 = Z21. This model is then transformed into an electrical equivalent circuit, which it is a lot easier to simulate, by transforming the mechanical to electrical parameters with the dual analog method. Figure 6 shows the equivalent electrical circuit after the transformer is eliminated, yielding a two-node circuit.
Cm 2
α RL
αEo
Ce/α
e
Rm
M2’
2
M1’
Figure 6. Electrical piezoelectric accelerometer’s model. The electrical and mechanical parameters can be obtained from the Bimorph manufacturer’s data or from experiment. In our case, these elements were obtained experimentally. Table No. 1 lists the electrical and mechanical parameters used in our simulation. Table 1. Electrical and mechanical parameters. Element Electrical capacitance Compliance Bimorph effective mass Support mass Mechanical resistance Electromechanical factor
Domain Electrical (Ce) = 128 nf (C2) = 345 µf (L1) = 1.56 mH (L2) = 57 mH (R) 0.33 Ω -3 (α) = 8.1x10 N/V
Mechanical e
-6
(Cm ) = 345x10 m/N -3 (M1’) = 1.56x10 Kg -3 (M2’) = 57x10 Kg (Rm) = 0.22 Kg/sec -3 (α) = 8.1x10 N/V
At this point, we are interested to find the relationship between the output voltage Eo and the free acceleration, Ao, of the object to which the accelerometer is attached. Analysis of the electrical model node equations, assuming RL large and Rm small, the transfer function Eo/Ao is as follows Eo Ao
=
[( e
α Cm + Ce / α
2
e Cm
)/ M1' − ω 2C me Ce / α 2 ]
(8)
3.1 Model Simulation Figure 7 shows the electrical simulation of our accelerometer. The predicted resonance frequency and gain of our accelerometer is 220 Hz and – 32dB, respectively. The flat range is from 1 to 130 Hz, at ± 3dB.
Figure 7. Accelerometer’s theoretical frequency response. 4. EXPERIMENTAL SET-UP AND MEASUREMENTS Figure 8 shows the block diagram of the experimental set-up. The prototype and a calibrated accelerometer were mounted on a mini-shaker. The characteristics of our prototype were compared with the standard accelerometer. A two-channel dynamic signal analyzer was used to determine the frequency response and sensitivity of our prototype.
Dynamic Signal Analyzer
Reference accelerometer
Oscilloscope
Tracking signal, white noise or sine sweep (0 – 500 Hz)
Accelerometer output voltage Charge amplifier
External mass M1
Motion
Power amplifier
Mini Shaker
Bimorph
Figure 8. Experimental set-up for determining the accelerometer frequency response. Figure 9 shows the experimental frequency response of both our prototype and the calibrated accelerometer. The resonance frequency of our accelerometer is 220 Hz, as expected. 0 Reference Bimorph
-5 -10 -15 -20 -25 -30 ) B d( e d ut i pl m A
-35 -40 -45 -50 -55 -60 -65 -70 -75 -80
5
55 Frequency (Hz)
105
155
205 255305
Figure 9. Bimorph and standard accelerometer’s frequency response. 5. DISCUSSION AND CONCLUSIONS According to the analytical model and experimental results, the gain and frequency response are in agreement. This simple 3-port device can then sense acceleration with high sensitivity at low frequencies. Adding an external mass in the port 1 can increase the sensitivity. However, the useful flat range of the accelerometer decreases. It turns out that our prototype is better by 25 dB compared to that of the reference. In other words, while the output voltage of the standard accelerometer is 1mV/1m/sec2, our device yields 18 mV/1m/sec2. Finally, this project had shown that Bimorph accelerometers make them attractive for low power, accurate, and sensitive applications. In addition, the low-cost and small size of this element, make possible their use in wide area seismic nets for earthquake and structural monitoring systems. 6. REFERENCES 1. J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 2nd ed. (American Institute of Physics, Woodbury, New York), (1997). 2. F. A. Fischer, Fundamentals of Electroacoustics, (Interscience Publishers, New York), (1955). 3. E. L. Hixson, Electromechanical Sensors and Actuators Class Notes (The University of Texas at Austin), (Spring 1998).
