# Investigating the Performance of a Fractal Ultrasonic Transducer Under Varying System Conditions

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## Abstract

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## 1. Introduction

## 2. Fractal Transducer Design

## 3. Mathematical Model for the Fractal Transducer

## 4. Investigating the Robustness of Transducer Performance

#### 4.1. Transducer Performance with Uncertainty in the Key Material Parameters

#### 4.2. Transducer Performance under Various Design Regimes

#### 4.3. Transducer Performance under Various Design Regimes with Uncertainty in Material Parameter Values

#### 4.4. Discussion of Transducer Performance

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

${\widehat{A}}^{(1)}$ | Matrix used to construct ${\widehat{G}}^{(1)}$ |

${A}_{r}$ | Cross-sectional area of each edge of the Sierpinski gasket transducer (m${}^{2}$) |

a | ${Z}_{P}/({Z}_{0}+{Z}_{P})$ (ratio of electrical impedances in the circuit (non-dimensional)) |

b | ${Z}_{0}{Z}_{P}/({Z}_{0}+{Z}_{P})$ (ratio of electrical impedances in the circuit (Ω)) |

${C}_{0}$ | The capacitance of the transducer (F) |

${c}_{44}$ | An element of the stiffness tensor of the piezoelectric material (N·m${}^{-2}$) |

${c}_{T}$ | The (piezoelectrically stiffened) wave velocity in the SG${}^{(n)}(3)$ lattice (m·s${}^{-1}$) |

${c}_{L}$ | Wave speed in the front load (m·s${}^{-1}$) |

${c}_{B}$ | Wave speed in the backing layer (m·s${}^{-1}$) |

${e}_{24}$ | An element of the piezoelectric tensor of the piezoelectric material (C·m${}^{-2}$ or N·V${}^{-1}$·m${}^{-1}$) |

f | The natural frequency |

${G}_{ji}^{(n)}$ | The Green’s transfer matrix given by Equation (1)–Equation (4) (non-dimensional) |

${\widehat{G}}^{(n)}$ | Recursive relations utilised for the renormalisation approach, given by Equation (7) and Equation (8) (non-dimensional) |

h | Length of each edge of the Sierpinski gasket transducer (m) |

${K}^{(n)}$ | Non-dimensionalised parameter given by Equation (12) |

L | Thickness of transducer (m) |

m | The vertex labelled $(N+1)/2$ |

$N={3}^{n}$ | The total number of vertices in the SG${}^{(n)}(3)$ lattice |

n | The fractal generation level |

q | Scaled frequency |

${Z}_{B}$ | Mechanical impedance of backing layer (N·s·m${}^{-1}$) |

${Z}_{L}$ | Mechanical impedance of the front load (N·s·m${}^{-1}$) |

${Z}_{T}$ | Mechanical impedance of the transducer (N·s·m${}^{-1}$) |

${Z}_{P}$ | Parallel circuit electrical impedance load (Ω) |

${Z}_{0}$ | Series circuit electrical impedance load (Ω) |

${Z}_{E}(f;n)$ | Electrical impedance of the SG${}^{(n)}(3)$ lattice given by Equation (11) (Ω) |

α | Non-dimensionalised parameter equal to $4+(8{q}^{2}/5)$ |

β | Non-dimensionalised parameter equal to $-2/3+11{q}^{2}/30$ |

${\gamma}_{j}$ | Non-dimensionalised parameter given by Equation (5) |

${\widehat{\gamma}}_{j}^{(n)}$ | Non-dimensionalised parameter equal to ${\eta}_{j}^{(n)}{\gamma}_{j}$ |

${\epsilon}_{11}$ | Element of the permittivity tensor of the piezoelectric material (F·m${}^{-1}$) |

ζ | ${e}_{24}/{\epsilon}_{11}$ (V·m${}^{-1}$) |

${\eta}_{j}^{(n)}$ | Non-dimensionalised parameter given by Equation (6) |

${\mu}_{T}$ | The (piezoelectrically stiffened) shear modulus (N·m${}^{-2}$) |

ξ | ${A}_{r}/h$ (m) |

${\rho}_{B}$ | Density of the backing layer (kg·m${}^{-3}$) |

${\rho}_{L}$ | Density of the front load (kg·m${}^{-3}$) |

${\rho}_{T}$ | The density of the piezoelectric material (kg·m${}^{-3}$) |

${\sigma}_{1}$ | Non-dimensionalised parameter given by Equation (13) |

${\sigma}_{2}$ | Non-dimensionalised parameter given by Equation (14) |

$\varphi (f;n)$ | The non-dimensionalised reception sensitivity given by Equation (10) |

$\psi (f;n)$ | The non-dimensionalised transmission sensitivity given by Equation (9) |

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**Figure 2.**Non-dimensionalised transmission sensitivity (Equation (9)) versus frequency for the $SG(3)$ lattice transducer at fractal generation level $n=3$ (dashed line). The non-dimensionalised transmission sensitivity of the standard (Euclidean) transducer is plotted for comparison (full line). Parameter values are given in Auld [29] for PZT-5H.

**Figure 3.**Non-dimensionalised reception sensitivity (Equation (10)) versus frequency for the $SG(3)$ lattice transducer at fractal generation level $n=3$ (dashed line). The non-dimensionalised reception sensitivity of the standard (Euclidean) transducer is plotted for comparison (full line). Parameter values are given in Auld [29] for PZT-5H.

**Figure 4.**Non-dimensionalised pulse-echo sensitivity (Equation (15)) versus frequency for the $SG(3)$ lattice transducer at fractal generation level $n=3$ (dashed line). The non-dimensionalised pulse-echo sensitivity of the standard (Euclidean) transducer is plotted for comparison (full line). Parameter values are given in Auld [29] for PZT-5H.

**Figure 5.**Comparison of the distribution of the maximum gain at various noise levels for the Sierpinski gasket lattice transducer at fractal generation level $n=3$ and the standard (Euclidean) transducer, as all key material parameters are subjected to a random perturbation: (

**a**) for the transmission sensitivity; (

**b**) for the reception sensitivity; and (

**c**) for the pulse-echo sensitivity. Parameter values are given in Auld [29] for PZT-5H.

**Figure 6.**Comparison of the distribution of the bandwidth at various noise levels for the Sierpinski gasket lattice transducer at fractal generation level $n=3$ and the standard (Euclidean) transducer, as all key material parameters are subjected to a random perturbation: (

**a**) for the transmission sensitivity; (

**b**) for the reception sensitivity; and (

**c**) for the pulse-echo sensitivity. Parameter values are given in Auld [29] for PZT-5H.

**Figure 7.**Comparison of the distribution of the maximum gain between the Sierpinski gasket lattice transducer at fractal generation level $n=3$ and the standard (Euclidean) transducer, as the key material parameters are subjected to a random perturbation: (

**a**) for the transmission sensitivity at a noise level of 30 dB; (

**b**) for the reception sensitivity at a noise level of 30 dB; (

**c**) for the pulse-echo sensitivity at a noise level of $-45$ dB. Parameter values are given in Auld [29] for PZT-5H.

**Figure 8.**Comparison of the distribution of the bandwidth between the Sierpinski gasket lattice transducer at fractal generation level $n=3$ and the standard (Euclidean) transducer, as the key material parameters are subjected to a random perturbation: (

**a**) for the transmission sensitivity at a noise level of 30 dB; (

**b**) for the reception sensitivity at a noise level of 30 dB; and (

**c**) for the pulse-echo sensitivity at a noise level of $-45$ dB. Parameter values are given in Auld [29] for PZT-5H.

**Figure 9.**Variation of the maximum gain for the transmission sensitivities calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 10.**Variation of the bandwidth (MHz) for the transmission sensitivities at a noise level of 30 dB, calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 11.**Variation of the maximum gain for the reception sensitivities calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 12.**Variation of the bandwith (kHz) for the reception sensitivities at a noise level of 30 dB, calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 13.**Variation of the maximum gain for the pulse-echo sensitivities calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 14.**Variation of the bandwith (kHz) for the pulse-echo sensitivities at a noise level of $-45$ dB, calculated over ranges of the design parameters ξ and L: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 15.**Coefficient of variation of the maximum gain for the transmission sensitivities at a noise level of 30 dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 16.**Coefficient of variation of the bandwidth for the transmission sensitivities at a noise level of 30 dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 17.**Coefficient of variation of the maximum gain for the reception sensitivities at a noise level of 30 dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 18.**Coefficient of variation of the bandwidth for the reception sensitivities at a noise level of 30 dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 19.**Coefficient of variation of the maximum gain for the pulse-echo sensitivities at a noise level of $-45$ dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Figure 20.**Coefficient of variation of the bandwidth for the pulse-echo sensitivities at a noise level of $-45$ dB over ranges of the design parameter values as all key material parameters are subjected to a random perturbation: (

**a**) the Sierpinski gasket lattice transducer at fractal generation level $n=3$; and (

**b**) the standard (Euclidean) transducer. Parameter values are given in Auld [29] for PZT-5H.

**Table 1.**Expected parameter values for PZT-5H (see Auld [29]), and geometrical parameters for the transducer design.

Parameter description | Symbol | Magnitude | Dimensions |
---|---|---|---|

Element of the stiffness tensor | ${c}_{44}$ | $2.3\times {10}^{10}$ | N·m${}^{-2}$ |

Element of the piezoelectric tensor | ${e}_{24}$ | 17 | C·m${}^{-2}$ |

Element of the permittivity tensor | ${\epsilon}_{11}$ | $1.51\times {10}^{-8}$ | C (V·m)${}^{-1}$ |

Density of the piezoelectric material | ${\rho}_{T}$ | 7500 | kg·m${}^{-3}$ |

Parallel electrical impedance load | ${Z}_{P}$ | 1000 | Ω |

Series electrical impedance load | ${Z}_{0}$ | 50 | Ω |

Length of transducer | L | 1 | mm |

Ratio of cross-sectional area to edge length | ξ | 0.4 | m |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Barlow, E.; Algehyne, E.A.; Mulholland, A.J.
Investigating the Performance of a Fractal Ultrasonic Transducer Under Varying System Conditions. *Symmetry* **2016**, *8*, 43.
https://doi.org/10.3390/sym8060043

**AMA Style**

Barlow E, Algehyne EA, Mulholland AJ.
Investigating the Performance of a Fractal Ultrasonic Transducer Under Varying System Conditions. *Symmetry*. 2016; 8(6):43.
https://doi.org/10.3390/sym8060043

**Chicago/Turabian Style**

Barlow, Euan, Ebrahem A. Algehyne, and Anthony J. Mulholland.
2016. "Investigating the Performance of a Fractal Ultrasonic Transducer Under Varying System Conditions" *Symmetry* 8, no. 6: 43.
https://doi.org/10.3390/sym8060043