Characterization of PZ27 and PZ52 Piezoceramics from Electrical Measurements

The characterization of ceramics is essential for the optimization of ultrasonic transducers. To do this we must determine the functional properties of ceramics which are: • speed of vibration of longitudinal waves • kt = Coupling coefficient: indicates the ability of the ceramic to transform electrical energy into mechanical energy • the dielectric constant • electrical and mechanical losses • acoustic impedance The electrical measurements allowed us to determine the functional properties of the ceramics available to us. We were able to refine the results thus obtained thanks to the digital simulator (KLM).


Introduction
This work aims to study the functional properties of piezoelectric ceramics for high intensity focused ultrasound applications (Souchon et al., 2003). Indeed, the latter are used in the health field for therapy applications such as the treatment of prostate cancer (Chapelon et al., 2000). In response to the needs of these new applications, new compositions must be developed by ceramic manufacturers. Some materials have recently been proposed, but improvements remain to be made. Thus, we received samples of PZ27 and PZ52 manufactured by the Danish company Meggitt Sensing System. First, by following the analytical approach, the electrical impedance will be calculated using Mason's model for piezoelectric plates. The calculation of the electrical and mechanical losses and their integration into the Mason model makes it possible to obtain a complex electrical impedance.
Electrical measurements are carried out and allow the determination of the piezoelastic properties.

Mathematical Model
The methods for characterizing active materials are mainly based on measuring electrical impedance as a function of the frequency of the piezoelectric resonator. Figure 1 shows the representation of a plate piezoelectric ceramic with two faces completely metalized. Without losses, the expression of electrical impedance of a plate piezolectric charged by two material, which mecanical impedance are Z 1 and Z 2 , in front and rear is: (Dieulesaint, Eugène, & Royer, 1996) When the piezoceramic faces are free, then Z 1 =Z 2 =0 (the forces are null on the faces), the electrical impedance becomes: The velocitty inside the piezoelectric element is : At the frequency a f . 2 , the tangent vanishes in the relation giving the electrical impedance. It then remains: We deduce the static capacity which is : and the dielectric constant is: A real piezoelectric element have electrical losses, e δ , and mecanical losses, m δ , grouped under the term of global losses, δ . If we integrated the losses in the model, the electrical impedance becaomes : There now appears a non-zero real part in the expression for electrical impedance.
If the mechanical and dielectric losses are low (which is the case for PZT or lead niobate type materials), an approximate expression of the electrical impedance can be made by a single development of the tangent in the vicinity of the antiresonance frequency f a (Loyau, 2004). It follows from this development that the width at half height Δf of the peak of the real part of the electrical impedance is directly proportional to the losses: with Q the quality factor.   Finally, the measurement of the density of the piezoelectric element will be made by measuring the geometric dimensions of the sample and the mass of the sample.

Experimental Set up
In order to measure the electrical impedance (real and imaginary parts) of a free piezoelectric resonator as a function of frequency, we have an Agilent 4395A impedance analyzer coupled with an impedance measurement kit. The piezoelectric element, which both sides are metallized, will be placed in a gripping system allowing electrical contact on each side with little mechanical influence. Under these conditions, it is assumed that the piezoelectric element behaves like a free resonator. The measuring device is shown in Figure 2. Finally, before taking the measurement, the overall system is calibrated by measuring the impedance of reference loads (standard load 50Ω, short circuit and open circuit). Thanks to an IEEE link cable, the measurements are transferred to a computer with a Labview acquisition program. These dats will be processed.

Results
To carry out these measurements, we have five (5) samples of PZ27 ceramic slides (parallelepipedal shapes) numbered 5 to 9 and two samples of PZ52 cylindrical shapes. These are ceramics from the Danish company FERROPERM.  The following figure (Figure 4) shows the real part and the imaginary part of the mesured electrical admittance and the method to extract the resonnance frequency, the antiresonance frequencies and the pass band frequencies.

Figure 4. Electrical impedance of the PZ27 ceramic
From these signals, we obtain the results which are presented in the tables below:  The results show that the materials have a good coupling coefficient and a correct acoustic impedance. They are therefore good candidates for applications in ultrasound transduction in therapy (Jaffe et al., 1954).

Operating Mode
The previous analytical calculations allow, from an electrical impedance measurement, to obtain the main characteristics of a piezoelectric element. It is possible to refine the method of determining the characteristics by a numerical calculation. The theoretical electrical impedance curves (real parts are imaginary) calculated numerically as a function of frequency are adjusted by modifications of the input parameters on the measured electrical impedance curves. Convergence occurs through the minimization of an index accounting for the difference between the theoretical and experimental curves. One initializes the calculation of convergence with the characteristics obtained by analytical method (Maréchal et al., 2007).

Presentation of Result
From the initial loss factor, the dielectric and mechanical losses are adjusted in order to have the same Quality factor Q. The piezoelastic constants must also be adjusted to correctly locate the theoretical resonances, which allow the determination of the piezoelastic properties. The piezoelastic tensor is then modified using the KLM (Krimholtz et al., 1970) model so that the theoretical admittance spectra are fitted to the experimental one with a simplex routine (Nelder & Mead, 1965;Zahara & Kao, 2009). The determination of the real part of the piezoelastic properties is then deduced therefrom.
The results of the adjustment are shown in Figure 5 on the real and imaginary parts of admittance and impedance. For the coupling coefficient and the speed of the waves we have almost the same values with 0.2%. On the other hand for the losses we have a notable difference which can be explained by the fact that we had trouble determining f2 (see Figure 5 and Figure 6) on the blue curve.

Conclusion
This work allowed us to determine the piezoelastic properties of the PZ27 and PZ52 ceramics from the company Meggitt Sensing Systems. We have a large variation in the losses and this is justified by the fact that the curves are not perfectly smooth.
The results obtained with the electrical measurements make it possible to predict the behavior of the ceramic after integration into systems such as ultrasonic transducers.
In further studies we plan to: • model of the heating: we will associate the ceramic with water on the front face and a backing on the rear face and give an equivalent model and calculate the temperature rise of the ceramic under these conditions.
• do laser measurements: we will measure the amplitude as a function of the excitation voltage then as a function of the number of cycles.
• model other types of ceramics such as ceramics with uniform porosity or with a porosity gradient between the two faces.