normal pi Over 2 EndFraction left-parenthesis 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction right-parenthesis tangent StartFraction normal pi Over 2 EndFraction StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction"/>
1.4.3 Mechanical Quality Factor
The mechanical Q (also referred to as Q) is the ratio of the reactance to the resistance in the series equivalent circuit representing the piezoelectric resonator, which is related to the sharpness of the resonance frequency. The mechanical QM can be calculated using the equation:
(1.15)
where fr is the resonance frequency, f1 and f2 are frequencies at −3 dB of the maximum admittance. The mechanical QM is also related to the electromechanical coupling factor k, following the equation:
(1.16)
where
| F r | Resonance frequency (Hz) |
| F a | Anti‐resonance frequency (Hz) |
| Z m | Impedance at Fr (ohm) |
| C 0 | Static capacitance (Farad) |
1.5 Issues for Measuring Piezoelectric Properties
1.5.1 Measurement of Direct Piezoelectric Coefficient Using the Berlincourt Method
One of the most crucial figures of merits characterizing a piezoelectric material is the piezoelectric charge constant, also called the direct piezoelectric coefficient. It reflects the internal generation of electrical charges resulting from an applied mechanical force, as previously mentioned. Basically, the higher the piezoelectric charge constant, the more active a piezoelectric material is. A fast and accurate evaluation of the direct piezoelectric coefficient can be realized by the Berlincourt method associated with a quasi‐static piezo d33‐meter [6, 28–31]. In this method, sample size or geometric shape becomes a factor that need not be strictly taken into account. Besides, the availability and convenient operation of a d33‐meter are obvious, which make it a predominantly used method in practice. However, the name of “Berlincourt method” is often mistaken by some people nowadays as being synonymous with the quasi‐static measurements of the direct piezoelectric coefficient. The latter more broadly refers to the methods operating at low or quasi‐static frequencies, and its basic principle of was proposed in “Piezoelectric Ceramics” by Jaffe et al. The name of “Berlincourt” actually derives from the researcher, Don Berlincourt, who devoted a lot of effort to the development of the initial commercial d33 apparatus based on the quasi‐static measurement principle [28].
Here, we consider a simple case of measuring the d33 value of a ceramic sample poled along the three‐direction (z) to elucidate the mathematical basis for the Berlincourt method. In the common case, the interaction between the mechanical and electrical behavior can be described by the equation d33 = [δD3/δT3]E, where D3 denotes electric displacement along the three‐direction (z) and T3 denotes applied stress also along the three‐direction (z). For the practical measurement of d33, this equation can be altered as d33 = [(Q/A)⋯(F/A)] = (Q/F), where F is applied force, A is the acting area, and Q is charge developed. It is obvious that d33 can be determined via measuring the charge induced by a certain force applied on the piezoelectric samples, while the measurement of the areas can be neglected as they cancel out. It should be noted that a constant electric field as fulfilled in the short‐circuit condition is the prerequisite of this measurement. To achieve this condition, a large capacitor across the Device Under Test (DUT) or a virtual‐ground amplifier is often embedded in the test system.
Based on the aforementioned consideration, one can easily realize that the d33 meter must include at least two parts: the force loading system for applying a small oscillating force, and the electronics for the circuit control and the charge measurement. As shown in Figure 1.5, the force loading system can further be divided to three parts, namely, contact probes, loading actuator, and reference sample [31]. The loading actuator is usually in the form of a loudspeaker type coil, which is cheap and easily controllable using electronic signals. The reference sample is used to monitor the applied force. It should be a piezoelectric material with a known piezoelectric coefficient and under the same force condition as the DUT. Thus, it is always put in the same loading line. PZT ceramics are usually served as the reference sample due to their high sensitivity. When the DUT is stimulated under the oscillating force with a certain frequency controlled by an amplified AC signal generated by the electronics, the corresponding charge is simultaneously measured by the electronics and the charge detected from the reference sample is converted to the actual force amplitude. Root Mean Squared (RMS) values of these signals are collected to determine the d33 value. It is worth nothing that some means of calibration must be conducted to give the correct value as the processing results from RMS signals are only proportional to the d33. Thus, reference calibration sample with a certified value is usually used. Finally, the electronics display a digital readout of the calculated d33 value.
Figure 1.5 Schematic illustration of the components in the force loading system.
Source: Modified from Cain [31].
The main advantage of the Berlincourt method lies in its simplicity. However, due to its simplicity, anyone can design their own measurement systems and no uniform standard about this method were established. We have found various commercial systems with different measurement performance. Though it is still reliable to compare the results measured in the same apparatus, large variabilities exist in the test results from different measurement systems. There are several factors controlling the accuracy and reliability of the results associated with the Berlincourt method. This issue might be briefly introduced by taking, for example, the magnitude and frequency of the applied AC load force. The magnitude of the force does not make much difference as long as if the piezoelectric sample behaves linearly within the stress range. Nonetheless, the increased magnitude of the force is expected to generate a larger charge signal, which helps increase the signal to noise ratio. It is thus better to set the force to at high levels as long as it can still keep the piezoelectric operating in the linear regime. For a typical Berlincourt type instrument, the frequency of the stimulus force usually ranges from 10 Hz