This article was downloaded by: [Professor kenji uchino] On: 22 October 2014, At: 12:04 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Ferroelectrics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gfer20
Methodology for Characterizing Loss Factors of Piezoelectric Ceramics a
a
ab
Yuan Zhuang , Seyit O. Ural & Kenji Uchino a
International Center for Actuators and Transducers, Pennsylvania State University, University Park, PA 16802, U.S.A. b
Office of Naval Research Global – Asia, Minato-ku, Tokyo 106-0032, Japan Published online: 20 Oct 2014.
To cite this article: Yuan Zhuang, Seyit O. Ural & Kenji Uchino (2014) Methodology for Characterizing Loss Factors of Piezoelectric Ceramics, Ferroelectrics, 470:1, 260-271, DOI: 10.1080/00150193.2014.923727 To link to this article: http://dx.doi.org/10.1080/00150193.2014.923727
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Ferroelectrics, 470:260–271, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 0015-0193 print / 1563-5112 online DOI: 10.1080/00150193.2014.923727
Methodology for Characterizing Loss Factors of Piezoelectric Ceramics YUAN ZHUANG,1,∗ SEYIT O. URAL,1 AND KENJI UCHINO1,2 1
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International Center for Actuators and Transducers, Pennsylvania State University, University Park, PA 16802, U.S.A. 2 Office of Naval Research Global – Asia, Minato-ku, Tokyo 106-0032, Japan The methodology to obtain the loss factors in piezoelectric ceramics is proposed in this paper. There are three hysteresis loss components for piezoelectric vibrators, i.e., dielectric, elastic and piezoelectric losses. The equations were derived about the relations between mechanical quality factors and all contributing loss factors by the complex analysis of the admittance/impedance expressions for specific piezoelectric vibrators. By characterizing mechanical quality factors and coupling coefficients in k31 , kt , k33 , kp , and k15 modes, 20 loss factors can be obtained for the piezoelectric ceramic with ∞mm or 6 mm crystal symmetry. Keywords Piezoelectric loss; high power characteristics; intensive/extensive losses; resonance/antiresonance
1. Introduction The key factor tothe miniaturization of piezoelectric devices is the power density, which is limited by material’s inherent losses that stem from the microscopic domain dynamics, resulting in the heat generation [1–3]. Therefore, to advance the power level of piezoelectric devices it is necessary to clarify the loss phenomenology and mechanism. Hysteresis losses in piezoelectrics are considered to have three types in general: dielectric, elastic, and piezoelectric losses [4–7]. Further each loss can be classified into intensive or extensive factor upon the boundary conditions [8, 9]. The dielectric and elastic loss factors are commonly reported by researchers and companies, while so far little attention has been paid to the piezoelectric loss factor [10]. However, relatively large piezoelectric loss factors were reported in the previous study [11]. Mechanical quality factors play a significant role in the loss study of piezoelectrics, and they are basically related to dielectric, elastic and piezoelectric loss factors. Besides, a higher quality factor at the antiresonance is usually observed in the PZT based piezoelectric materials, in comparison with the resonance quality factor [12–14]. However, the previous theory without considering the piezoelectric loss factor could not explain the deviation of the resonance quality factor QA and antiresonance quality factor QB explicitly. IEEE Std. only provided the method to derive QA based on the equivalent circuit, and assumed that the resonance quality factor is equal to the antiresonance quality factor from a traditional thought [15]. Received in final form August 1, 2013. ∗ Corresponding author; E-mail:
[email protected]
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Loss Characterization Methodology
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In this paper, the equations for piezoelectric quality factors were derived with regard to three loss factors and other material properties. The difference of QA and QB was explained by the results. Within the study we focused on piezoelectric ceramics with ∞mm or 6mm crystal symmetry, exemplified by the conventional lead zirconate titanate (PZT) ceramics with the uniform distribution of fine grains. To cover 20 material property parameters, k31 , kt , k33 , kp , and k15 vibration modes were analyzed. Then the methodology to derive loss factors was proposed, based on the theoretical equations. By characterizing quality factors and other parameters 20 loss factors can be obtained. The loss characterization methodology was applied on PZT ceramic APC 850. All the real and imaginary material properties were obtained, and the orientation dependence of the loss factors was accordingly discussed.
2. Derivations of Quality Factors Complex parameters are integrated to express the hysteresis losses in piezoelectrics [16]. We use tanδ , tanφ and tanθ to represent intensive dielectric, elastic and piezoelectric loss factors, respectively. The extensive loss factors are given by corresponding notations without prime. The definitions are given by εT ∗ = εT (1 − j tan δ ), s
E∗
= s (1 − j tan φ ), E
∗
(1) (2)
d = d(1 − j tan θ ),
(3)
β S∗ = β S (1 + j tan δ),
(4)
= c (1 + j tan φ),
(5)
c
D∗
D
h∗ = h(1 + j tan θ ).
(6)
Here j is the imaginary notation, εT the dielectric constant under constant stress T, β the inverse dielectric constant under constant strain S, sE the elastic compliance under constant electric field E, cD the elastic stiffness under constant electric displacement D, d the piezoelectric constant, and h the inverse piezoelectric charge constant. From the physics viewpoint, an intensive property is a physical property of a system that does not depend on the system size or the amount of material in the system, while the extensive property is directly proportional to the size or amount. In addition, state variables are extensive and field variables are intensive [17, 18]. As for the loss factors, intensive loss corresponds to the boundary conditions of constant T or E, while extensive loss is attributed to the boundary conditions of constant S or D. Here T and E are intensive parameters, and S and D are extensive properties. Note that the phenomenological equations only hold when the piezoelectric sample works in the linear region and the loss is treated as a perturbation. In practice, the theoretical equations derived in this way are accurate for the case in which the loss factors are less than 0.1. Then we derived the mechanical quality factors of piezoelectric ceramics in k31 , kt , k33 , kp , and k15 vibration modes. Each QA or QB equation is obtained from the complex impedance/admittance expression of the specific mode based on the 3dB definition as shown in Fig. 1, utilizing the first order approximation. The derivation method and details can be found in our previous publications [19, 20]. S
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Figure 1. Definitions of quality factors at (a) resonance and (b) antiresonance.
The geometries for the vibration modes are shown in Fig. 2. There are dimension requirements for each mode, according to the standard [15]. Notice that there are two configurations of k15 shear samples. The thickness shear mode (L>> t) corresponds to the constant induction condition, and the length shear mode (L
