Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales

Algehyne, Ebrahem A. and Mulholland, Anthony J. (2019) Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales. IMA Journal of Applied Mathematics, 84 (3). pp. 605-631. ISSN 1464-3634 (

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The transmission and reception sensitivities of most piezoelectric ultrasonic transducers are enhanced by their geometrical structures. This structure is normally a regular, periodic one with one principal length scale which, due to the resonant nature of the devices, determines the central operating frequency. There is engineering interest in building wide bandwidth devices, and so it follows that in their design, resonators that have a range of length scales should be used. This paper describes a mathematical model of a fractal ultrasound transducer whose piezoelectric components span a range of length scales. There have been many previous studies of wave propagation in the Sierpinski gasket but this paper is the first to study its complement. This is a critically important mathematical development as the complement is formed from a broad distribution of triangle sizes whereas the Sierpinski gasket is formed from triangles of equal size. Within this structure, the electrical and mechanical fields fluctuate in tune with the time dependent displacement of these substructures. A new set of basis functions is developed that allow us to express this displacement as part of a finite element methodology. A renormalisation approach is then used to develop a recursion scheme that analytically describes the key components from the discrete matrices that arise. Expressions for the transducer's operational characteristics are then derived and analysed as a function of the driving frequency. It transpires that the fractal device has a significantly higher reception sensitivity (18 dB) and a significantly wider bandwidth (3 MHz) than an equivalent Euclidean (standard) device.