0885–3010/$25.00 © 2010 IEEE 969 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, . 57, . 4, APRIL 2010 Abstract—The radiation impedance of a capacitive micro- machined ultrasonic transducer (cMUT) with a circular mem- brane is calculated analytically using its velocity proﬁle for the frequencies up to its parallel resonance frequency for both the immersion and the airborne applications. The results are veriﬁed by ﬁnite element simulations. The work is extended to calculate the radiation impedance of an array of cMUT cells positioned in a hexagonal pattern. A higher radiation resis- tance improves the bandwidth as well as the eﬃciency of the cMUT. The radiation resistance is determined to be a strong function of the cell spacing. It is shown that a center-to-center cell spacing of 1.25 wavelengths maximizes the radiation resis- tance, if the membranes are not too thin. It is also found that excitation of nonsymmetric modes may reduce the radiation resistance in immersion applications. I. I C  micromachined ultrasonic transducers (cMUTs) oﬀer wider bandwidth in air [1]–[4] and in water [5]–[8] compared with their piezoelectric alterna- tives due to their low mechanical impedances. The limit for the bandwidth is the parallel resonance frequency of the cMUT membrane in water, whereas the mechanical impedance of the membrane limits the bandwidth in air. In this work, the acoustic loading on the circular cMUT membranes is investigated by calculating their radiation impedances. The mechanical impedance of a cMUT membrane in vacuum is well studied [9]. It shows successive series and parallel resonances, where force and velocity become zero, respectively [10]. When a cMUT is immersed in water, the acoustic loading on the cell is high and results in a wide bandwidth. All mechanical resonance frequencies shift to lower values because of the imaginary part of the radiation impedance. If a cMUT is used in air, the radiation imped- ance is rather low, and the bandwidth is limited by the mechanical Q of the membrane. It is therefore preferable to increase the radiation resistance to get a higher band- width in airborne applications. In addition, for the same membrane motion, a higher acoustic power is delivered to the medium, if the radiation resistance is higher. Hence, a higher radiation resistance is desirable to be able to trans- mit more power, because the gap limits the maximum al- lowable membrane motion. The eﬃciency of a transducer is deﬁned as the ratio of the power radiated to the medium to the power input to the transducer [11]. The loss in a cMUT due to the electrical resistive eﬀects and the mechanical power lost to the sub- strate can be represented as a series resistance [1]. Hence, the eﬃciency will increase if the radiation resistance in- creases in both airborne and immersion cMUTs, because a smaller portion of the energy will be dissipated on the loss mechanisms such as the coupling into the substrate. There are several approaches to model the radiation impedance of the cMUT membrane. In [12], the radiation impedance is modeled using an equal size piston radiator. In [13], an equivalent piston radiator with the appropriate boundary conditions is deﬁned and its radiation impedance is used. In [14], the radiation impedance of an array is mod- eled with lumped circuit elements. In [15], the radiation im- pedance is calculated by subtracting the mechanical imped- ance of the membrane from the input mechanical impedance as computed by a ﬁnite element simulation. In [16], cMUT is modeled with a modal expansion-based method, and the radiation impedance is calculated using that method. Caro- nti et al. [17] calculated the radiation impedance of an array of cells performing ﬁnite element simulations with a focus on the acoustic coupling between the cells. The radiation impedance of an array of cMUT cells is not well known. In this work, the radiation impedance of an array of cMUT cells with circular membranes is pre- sented. First, the radiation impedance of a single cMUT cell is calculated using its velocity proﬁle. Then, the radia- tion impedance of array of cMUT cells is calculated from analytical expressions and compared with those found from ﬁnite element simulations. II. M B   C MUT M A. Finite Element Method (FEM) Simulations FEM simulations are performed using ANSYS 1 (AN- SYS Inc., Canonsburg, PA) in water [18]–[20] to calculate Radiation Impedance of an Array of Circular Capacitive Micromachined Ultrasonic Transducers Muhammed N. Senlik, Student Member, IEEE, Selim Olcum, Student Member, IEEE, Hayrettin Köymen, Senior Member, IEEE, and Abdullah Atalar, Fellow, IEEE Manuscript received January 30, 2009; accepted December 9, 2009. This work is supported in part by the Turkish Scientiﬁc and Research Council (TUBITAK) under project grants 105E23 and 107T921. S. Ol- cum acknowledges the support of TUBITAK and ASELSAN for their Ph.D. scholarship programs. A. Atalar thanks TUBA for the research support. The authors are with the Electrical and Electronics Engineer- ing Department, Bilkent University, Ankara, Turkey (e-mail: niyazi@ ee.bilkent.edu.tr). Digital Object Identiﬁer 10.1109/TUFFC.2010.1501 1 The membrane, the ﬂuid, and the absorbing boundary are modeled using PLANE42, FLUID29, and FLUID129 elements, respectively.