# Applied Mathematics and Statistics Research Unit

## Piezoelectric Ultrasonic Transducers

Ultrasonic transducers are devices that convert electrical energy into mechanical vibration and conversely can convert mechanical energy into an electrical signal [1]. These devices can be used to interrogate a medium by emitting a wave (electrical to mechanical) and then listening to the same wave after it has traversed the medium (mechanical to electrical). Transducers occur in many natural forms, particularly in biological species such as bats, dolphins and cockroaches. Manufactured transducers have a very regular geometry with a single length scale whereas natural systems normally have a complex geometry, often with resonators over a range of length scales [2,3,4,5,6,7]. Therefore, it is of interest to assess the benefits of designing a transducer with a complex structure. Fractals are structures whose geometrical components cover a range of length scales and hence this article will consider ultrasound transducers with fractal geometry [8].

Piezoelectric ultrasonic transducers are typically composed of a two phase material; a piezoelectric material and a damping phase. Traditionally the geometry of these phases are manufactured to be in regular `checker board’ type arrangements. As these are resonant devices then the single length scale arising from this structure dictates the central frequency of its thickness mode operation. In order to extend the transmission and reception sensitivity of the device over a wider bandwidth it would seem natural therefore to consider a heterogeneous structure that contains many length scales. Indeed the broadband resonators found in nature and in the development of musical instruments rely on this principle. The aim of this article therefore was to derive a mathematical model to predict the dynamics of a fractal ultrasound transducer; the fractal in this case being the Sierpinski gasket. The use of a fractal structure has the additional advantage of being amenable to a renormalisation analysis. By discretising the Laplacian in this fractal lattice the renormalisation method was used to determine the displacement of each vertex in the structure as a function of frequency. By coupling the lattice to an electrical and a mechanical load, and applying appropriate interface and boundary conditions, expressions for the electrical impedance and the transmission and reception sensitivities were derived. These expressions were then compared to their Euclidean counterparts that have been derived for the homogeneous case.

The results [9,10] show that the fractal transducer is, as anticipated, resonant at many more frequencies than the homogeneous transducer. Unfortunately, in transmission mode, the homogeneous transducer outperforms the fractal transducer and there are additional concerns over the uniformity of the structure in producing a coherent bean profile. However, in reception mode, the fractal transducer outperforms the homogeneous transducer at almost all frequencies. This encouraging evidence may be directly related to the range of length scales which the fractal transducer possesses.

The convergence of the device performance as the fractal generation level is increased was also considered. It was seen that, in both transmission and reception modes, the outputs converge by generation level n=45. Pre-fractal transducers of this generation level have manufacturing difficulties but the results show that at very low generation levels, that are within manufacturing tolerances, there is a marked improvement in reception sensitivity. The fractal used in this research was the Sierpinski gasket, however other fractal structures exist and could be investigated in a similar way as candidate transducers. Furthermore, it may be possible to tackle the inverse problem of designing a specific fractal structure for a desired set of transducer operating characteristics.

Ph.D. student, Sara Koubayssi, is currently researching other self-similar structures which could be used in a mannar similar to that described above. Analysis will be carried out to ascertain which structures provide the strongest ultrasonic receivers and transmitters.

References:

[1] J. Yang, The Mechanics of Piezoelectric Structures, World Scientific, Singapore, 2006.

[2] R. Müller, H. Lu, S. Zhang, and H. Peremans, A helical biosonar scanning pattern in the chinese noctule Nycatalus plancyi, J. Acoust. Soc. Am., 119(6):4083—4092, 2006.

[3] R. Müller, A numerical study of the role of the tragus in the big brown bat. J. Acoust. Soc. Am., 116(6):3701—3712, 2004.

[4] R. N. Miles and R. R. Hoy, The development of a biologically-inspired directional microphone for hearing aids, \em Audiol Neurootol., 11(2):86—94, 2006.

[5] A. M. Chiselev, L. Moraru, and A. Gogu., Localization of an object using a bat model inspired from biology, Romanian. J. Biophys., 19(4):251—258, 2009.

[6] D. F. Eberl, R. W. Hardy, and M. J. Kernan, Genetically similar transduction mechanisms for touch and hearing in Drosophila, J. Neurosci., 20(16):5981—5988, 2000.

[7] B. Nadrowski, J. T. Albert, and M. C. Göpfert, Transducer-based force generation explains active process in Drosophilia hearing, Curr. Biol., 18:1365—1372, 2008.

[8] D. Robert and M. C. Göpfert, Novel schemes for hearing and orientation in insects, Curr. Opin. Neurobiol., 12:715—720, 2002.

[9] A. J. Mulholland and A. J. Walker, Piezoelectric Ultrasonic Transducers with Fractal Geometry, Fractals, 2011, 19(4), 469-479.

[10] A. J. Mulholland, J. W. Mackersie, R. L. O’Leary, A. Gachagan, A. J. Walker, and N. Ramadas, The use of Fractal Geometry in the Design of Piezoelectric Ultrasonic Transducers, Proceedings of the 2011 IEEE International Ultrasonics Symposium , Orlando, Florida, pp1559-1562, (2011), DOI: 10.1109/ULTSYM.2011.0387