# Estimation of Acoustic Power Output from Electrical Impedance Measurements

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Overview of Current Methods

#### 1.2. Equivalent Circuit Models for Transducers with Reference to the Current Work

## 2. Theory

#### 2.1. Two-Port Transducer Model

#### 2.2. Estimation of Two-Port Parameters and Power

**x**(Equation (5)). However, using the assumption of reciprocity, generally true for circuits containing passive elements only and shown to be true for the KLM model [7], ${Z}_{12}={Z}_{21}$. During the calculation of the impedance parameters, the square root of ${Z}_{12}{Z}_{21}$ is taken, which gives a positive real impedance value for both ${Z}_{12}$ and ${Z}_{21}$. Since there are three unknown terms remaining in

**x**, at any given frequency, measurements of the input impedance ${Z}_{in}$ using at least three different acoustic loads ${Z}_{L}$ allows an estimate of the parameter vector $\mathbf{x}$ to be obtained using matrix inversion. The equation system will have a unique solution as long as the impedance properties of the three loads are different.

## 3. Materials and Methods

#### 3.1. Transducers Used for the Measurements

^{rd}harmonic frequencies: H-102, SN: B-022 (1.060/3.190 MHz) and H-107, SN: 031 (0.5/1.7 MHz). The fundamental and 3

^{rd}harmonic bandwidths, as defined by the manufacturer, down to -3dB normalized to a perfect 50 $\mathsf{\Omega}$ match, were 640/700 kHz (H-102) and 340/80 kHz (H-107). The H-102 and H-107 transducers had a rectangular and circular cutout, respectively, modifying their surface areas from 32.2 to 26.5 cm${}^{2}$ and 32.2 to 30.8 cm${}^{2}$, respectively (Figures 4a and 5a).

#### 3.2. Electrical Impedance Measurements

#### 3.3. Acoustic Characterization

#### 3.4. Measurements Validating the Range of Linearity for the Model

## 4. Results

#### 4.1. Comparison of Estimated and Measured Acoustic Powers

#### 4.2. Linearity of the Model

## 5. Discussion

#### 5.1. Interpretation of Transducer-Specific Phenomena Affecting Measurements

#### 5.2. Scaling of the Results

#### 5.3. Potential Advantages of the Proposed Method

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HIFU | High-intensity focused ultrasound |

KLM model | Equivalent circuit model of piezoelectric ultrasound transducers, named after R. Krimholtz, D. A. Leedom and G. L. Matthaei |

## Appendix A. Derivation of Equation (6) from Equations (1) and (2)

## Appendix B. Calculation of the Two-Port Network Parameters

## Appendix C. The Validity of Short-Pulse Measurements for HIFU Transducers

**Table A1.**Comparison of transfer function ratios for the H-107 (SN: 031) transducer system driven with short versus long pulses, in both driving frequencies of the transducer.

Drive Band (MHz) | Pulse Lengths (cycles) | Transfer Function Ratio, 200 kHz Band | Transfer Function Ratio, at Centre Frequency |
---|---|---|---|

“Fundamental” | 2, 12 | 0.994: 0.4–0.6 MHz | 0.998: 0.50 MHz |

“3rd Harmonic” | 2, 40 | 0.997: 1.6–1.8 MHz | 0.997: 1.70 MHz |

## Appendix D. Scaling of the H-107 Acoustic Measurements after Repetition

## Appendix E. Uncertainty Estimation of the Impedance Measurement-Based Method

## References

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**Figure 1.**Schematic of a single transducer element system, showing electrical connections, back and front acoustic loading, and electrical matching as an optional part of the network. ${V}_{1}$ and ${I}_{1}$ stand for the voltage and current at the electrical port of the transducer, while ${V}_{2}$ and ${I}_{2}$ represent the acoustic pressure and particle velocity of the front acoustic load, seen as a voltage and current, respectively, in the equivalent circuit model.

**Figure 2.**Schematic of the two-port network model with impedance parameters ${Z}_{11},{Z}_{12},{Z}_{21},{Z}_{22}$. ${Z}_{L}$ is the load impedance at port 2. The impedance measured at port 1 is ${Z}_{in}$.

**Figure 3.**Schematic of the (

**a**) electrical impedance measurements (Section 3.2) and of the (

**b**) acoustic characterization (pressure measurements from which acoustic power was calculated; see Section 3.3).

**Figure 4.**Comparison of electrically estimated (from impedance “Z” measurements) and acoustically measured (estimated from acoustic pressure “p” field measurements) power outputs (1 V peak drive voltage) of the H-102 (SN: B-022) transducer. (

**a**): Transducer surface schematic with the cutout. (

**b**,

**c**): Total (electric and acoustic) power and estimated and measured acoustic power. (

**d**,

**e**): Estimated and measured efficiency. The dotted vertical lines indicate the centre frequency defined by the manufacturer at the fundamental and third harmonic resonances of the transducer.

**Figure 5.**Comparison of electrically estimated (from impedance “Z” measurements) and acoustically measured (estimated from acoustic pressure “p” field measurements) power outputs (1 V peak drive voltage) of the H-107 (SN: 031) transducer. (

**a**): Transducer surface schematic with the cutout. (

**b**,

**c**): Total (electric and acoustic) power and estimated and measured acoustic power. (

**d**,

**e**): Estimated and measured efficiency. The dotted vertical lines indicate the centre frequency defined by the manufacturer at the fundamental and third harmonic resonances of the transducer.

**Figure 6.**Validation of the small-signal electrical impedance measurement results to higher driver voltages (resulting in higher pressures and powers). (

**a**) Focal pressure, (

**b**) electrical impedance, and (

**c**) radiated power measurements as a function of drive voltage in the range of 4–80 V showing the linearity of the pressure-voltage correlation, in the case of the third harmonic band (∼1.7 MHz) of the H-107 (SN: 031) transducer.

© 2020 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/).

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**MDPI and ACS Style**

Csány, G.; Gray, M.D.; Gyöngy, M. Estimation of Acoustic Power Output from Electrical Impedance Measurements. *Acoustics* **2020**, *2*, 37-50.
https://doi.org/10.3390/acoustics2010004

**AMA Style**

Csány G, Gray MD, Gyöngy M. Estimation of Acoustic Power Output from Electrical Impedance Measurements. *Acoustics*. 2020; 2(1):37-50.
https://doi.org/10.3390/acoustics2010004

**Chicago/Turabian Style**

Csány, Gergely, Michael D. Gray, and Miklós Gyöngy. 2020. "Estimation of Acoustic Power Output from Electrical Impedance Measurements" *Acoustics* 2, no. 1: 37-50.
https://doi.org/10.3390/acoustics2010004