Model for Radial Modes in a Thin Piezoelectric Annular Array He ´ctor CALA ´ S, Eduardo MORENO 1 , Jose ´ Antonio EIRAS 2 , Arturo VERA  , Roberto MUN ˜ OZ, and Lorenzo LEIJA Seccio ´n de Bioelectro ´nica, CINVESTAV, Av. I.P.N. 2508 Col. San Pedro Zacatenco, C.P. 07300, Me ´xico D.F., Mexico 1 Departamento de Fı ´sica Aplicada, Instituto de Ciberne ´tica, Matema ´tica y Fı ´sica, Calle D 551, Vedado, Cuba 2 Departamento de Fı ´sica, Grupo de Cera ´micas Ferroele ´ctricas, UFSCar. Rod. Washington Luis, km. 235, 13.565-905 San Carlos-SP, Brazil (Received May 15, 2008; accepted July 5, 2008; published online October 17, 2008) Piezoelectric arrays with different connectivities are widely used in actuators, transformers, and special ultrasonic transducers. In this study, the global matrix method is presented to describe the resonance radial modes in thin piezoelectric annular arrays with an arbitrary number of elements. Each annular element of the arrays is modeled as a three-port system with one electrical and two mechanical ports. The cored matrix model for radial modes, in a singular ring, is presented in the form of a translation matrix. The dynamic behavior of two annular arrays, a transformer and a Bessel transducer, is simulated assuming that the major surfaces of each element are stress-free, while the laterals are matched with neighbor elements. Thus, the inner and outer lateral surfaces of the system are loaded with an external medium. The simulated and experimental results are in good agreement when the thickness effects are neglected. [DOI: 10.1143/JJAP.47.8057] KEYWORDS: piezoelectric, radial modes, annular array, global matrix method 1. Introduction Transducers using piezoelectric arrays with annular configurations are extensively studied nowadays. 1–5) Their use in medical and industrial applications, as in the case of a linear transducer, is supported by the attributes of the control of the radiation field 1,4) or vibration modes. 5–7) In these devices, it is important to know the vibrational character- istics of radial modes, particularly those that affect the behavior of the device. Moreover, if the width and height of the array elements are similar, the radial modes will be coupled to the designed extensional resonance. 8) This array has negative consequences over the broadband and working frequency of the desired array. 9) Another problem is the cross coupling that can occur in this system. Therefore, a study of radial modes, with analytical theoretical models or numerical approaches, is required. One method of solving this problem numerically is to use the finite element method (FEM). Eiras et al. 4) and Li et al. 3) applied this technique in order to analyze radial modes in a Bessel transducer and in a piezoelectric ring transformer, respectively. Many authors have conducted theoretical studies of radial modes for piezoelectric disks and rings with homogeneous polarization. Berlincourt 10) proposed a simple model for a ring element with a small width. In 1973, Meitzler et al. 11) presented a model for a disk and introduced a new radial mode coupling factor. A volumetric stress-free model was developed by Brissaud 12) using only the boundary conditions on the external circumference of the two planar surfaces. Iula et al., 13) in 1996, proposed a two-dimensional (2D) model using a matrix description to predict radial modes for a thin ring. Matrix formulations have been used for thickness mode description in multilayer piezoelectric transducers. 14–16) In this contribution, a matrix method for the theoretical study of radial modes in piezoelectric annular array transducers is presented. The method is supported by the global matrix 17–19) method and a new translation matrix representa- tion 14) for piezoelectric rings. The system was considered to be multiple sections acoustically connected in cascade 20,21) and electrically connected in parallel. This type of approach for interconnected sections was developed by Smith et al. 22) for surface acoustic wave interdigital transducers; here, the behavior of the unit section corresponding to a pair of electrodes was derived by analogy with Mason’s equivalent circuit for bulk-wave transducers. 23) In our case, each section is studied using a matrix formalism. This method is also justified because it gives a good insight into the physical interpretation, it has numerical robustness, and it also permits the user to observe the behavior of each component of the system. In order to verify the model, the results obtained for two piezoelectric annular array transducers are shown and compared with those of the simulation. The experimental and theoretical results are in good agreement. 2. Radial Mode Vibration in Thin Piezoelectric Ring Figure 1 shows a schematic representation of the j-th ring inside an array, which is associated with a three-port model. Analysis will be made in cylindrical coordinates r, z, and , considering the origin to be the center of the ring and the polarization direction to be parallel to the z-axis. The thickness of the ring 2b is considered small in comparison to its external radius r j (2b  r j ); thus, any thickness deforma- tions can be neglected; however, radial deformations must Fig. 1. Three-port model for the j-th ring of the array.  E-mail address: arvera@cinvestav.mx Japanese Journal of Applied Physics Vol. 47, No. 10, 2008, pp. 8057–8064 #2008 The Japan Society of Applied Physics 8057