ORIGINAL ARTICLE

Mathematical Modeling and Analytic Characterization of an Ultrasound Proximity Sensor

Ing. Javier A. León-Martínez^{*  1} 

Dr.C. Antihus A. Hernández-Gómez¹ 

Dr.C. Ciro E. Iglesias-Coronel¹ 

¹Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba

ABSTRACT

This research work is about mathematical modeling and analytic characterization of a proximity sensor prototype based on ultrasonic acoustic echolocation with thermal compensation. The analytic characterization of the propagation medium behavior under the acoustic disturbance produced by the sensor is presented, starting from a correlation function fitted to the real excitation signal generated by it. The influence of the catoptrics conditions of the separation boundary between the air and a set of materials in echo intensity is also referred. Likewise, the sensor response to the echo (considering the worst catoptrics conditions) is analytically characterized.

Keywords: ultrasound sensor; acoustic excitation; acoustic reflectivity coefficient; echo intensity

INTRODUCTION

The automated measurement of proximity/distance is indispensable to face the growing complexity of a great number of productive systems. In this way, the sensors based on ultrasonic acoustic echolocation stand out. A diverse set of authors like ^{Siemens (2008)}, ^{Gómez & López (2009)}, ^{Montoya (2013)}, ^{Domínguez (2014)}, ^{Moreno (2016)} and ^{Bermudez (2017)}, mention that these systems have potential applications in food storage, control of fuel consumption, measurement of structural parameters in roadways networks and determination of water volume stored in tanks, dams and wells. Similarly, they are applied in specialized instruments, robotics, automation of agricultural and agroindustrial processes and many others.

These systems are viable due to some advantages in comparison with other methods of automated measurement of proximity/distance (^{Siemens 2008}; ^{Cuamatzi et al., 2010} and ^{Kentish, 2017}). Among them, high immunity to the mechanical vibrations, high immunity to adverse work conditions (environmental noise, dust, gases, others), measurement range from tenths of centimeters until meters and comparatively low cost, are relevant.

Likewise, the mathematical modeling of measurement systems has a very important role in the characterization, design and simulation of systems of automatic control of processes (^{Placko, 2006} and ^{Stephan, 2011}). For that reason this paper is about the mathematical modeling and the analytic characterization of a proximity sensor prototype based on ultrasonic acoustic echolocation with thermal compensation (see Figure 1), whose fundamentals and design have been treated in precedent works (^{León et al., 2018}, ²⁰¹⁹).

FIGURE 1 Ultrasound proximity sensor. 

In this paper, the analytic characterization of the propagation medium behavior under the acoustic disturbance produced by the sensor is presented. The influence of the catoptrics conditions of the separation limit in echo intensity is also shown. Likewise, the sensor response to the echo is analytically characterized, considering the worst catoptrics conditions studied.

METHODS

Theoretical Considerations

Acoustic Excitation under the Sensor Performance Conditions

The active element of the 10CK40T transducer used in the sensor, is a quartz piezoelectric ultrasonic buzzer. This acts as a peculiar type of filter on the pulses signal generated by the excitation subsystem of the sensor (^{León et al., 2018}). For that reason, the real excitation signal does not have a square shape (see blue curve in Figure 2). Therefore, in order to facilitate the characterization of its behavior, it would be convenient to fit it to an analytic correlation function, from recording the real excitation signal (^{Zilesny, 2011}). Then, it is convenient to use the following function:

$V_{exc}\left(t\right)=a\left[sin\left(2\pi f_{exc}t+c\right)+b\bullet sin\left(6\pi f_{exc}t+c\right)\right], V$ (1)

being:

$a$ :	Amplitude of the main harmonic component of $V_{exc}\left(t\right)$ function, $V$
$b$ :	Non-dimensional relationship among the amplitudes of the main and secondary harmonic components respectively
$c$ :	Phase of $V_{exc}\left(t\right)$ function, $rad$
$f_{exc}$ :	Nominal frequency of the transducer, $Hz$
$t$ :	Time, $s$

The parameters $a$ , $b$ and $c$ are the correlation coefficients for the fitting of the function (1) to the real signal generated by the sensor for the transducer excitation.

The wave function that corresponds to a spherical wave front ^{Crawford (1968)}; ^{Young & Freedman (2009)}; ^{Ginsberg (2018)}, generated according to the signal described by the equation (1), is determined for:

${\psi }_{exc}\left(r,t\right)=A_{0exc}\frac{r_{0exc}}{r}\left[sin\left(2\pi f_{exc}t-k_{exc}r+c\right)+\cdots \right.$

$\left.\cdots +b\bullet sin\left(6\pi f_{exc}t-{3k}_{exc}r+c\right)\right], m$ (2)

where:

$r$ :	Wave propagation distance in the instant of time $t$ , $m$ ;
$r_{0exc}$ :	Radio of the transducer emission surface, $m$ .

The wave number of the main harmonic component in the wave function (2), is determined according to:

$k_{exc}=\frac{2\pi f_{exc}}{v_s}, rad/m$ (3)

being:

$v_s$ :

Sound speed in the air, $m/s$ .

Thus, the oscillations amplitude on the emission surface of the transducer will be:

$A_{0exc}=\frac{1}{\pi f_{exc}}\sqrt{\frac{I_{0exc}}{2{\rho }_{air}{v}_s}}, m$ (4)

where:

${\rho }_{air}$ :

Density of the air, $kg/m^3$ .

Considering the sensor design conditions, the wave intensity on the emission surface of the transducer can be determined according to:

$I_{0exc}=I_{ref}{\bullet 10}^{\left(\frac{{\beta }_{max}}{10}\right)}\bullet \frac{2\bullet max\left[V_{exc}\left(t\right)\right]}{V_{ppmax}}, W/m^2$ (5)

being:

$I_{ref}$ :	Reference sound intensity, ${10}^{-12}W/m^2$ .
${\beta }_{max}$ :	Maximum sound pressure level of the transducer, $dB$ ;
$V_{ppmax}$ :	Maximum input voltage (peak to peak) of the transducer, $V$ .

Dependence between the Sensor Sensibility and the Catoptrics Properties of the Separation Limit

The elastic waves behavior in the separation limit between different densities propagation media, is characterized by the separation in two new wave fronts (^{Yavorski & Pinski, 1983}; ^{Young & Freedman, 2009}). They are known as reflected (echo) and refracted wave fronts, respectively.

The catoptrics properties characterize the wave reflection capacity in the limit of separation between two propagation media. Reflectivity coefficient has been defined as the relation between the intensities of the reflected and incident waves, quantifying, in this way, the separation limit catoptrics properties (^{Crawford, 1968a}; ^{Yavorski & Pinski, 1983}). In the particular case of a sonic wave traveling by the air, the acoustic reflectivity coefficient for the associate separation limit between the air and another material, is calculated according to:

$R_n={\left(\frac{{\rho }_{air}{v}_s-{\rho }_n{v}_n}{{\rho }_{air}{v}_s+{\rho }_n{v}_n}\right)}^2$ (6)

where:

$n$ :	Numeric identification associated to the corresponding material (see Table 1);
${\rho }_n$ :	Density of the corresponding material, $kg/m^3$
$v_n$ :	Sound speed in the corresponding material, $m/s$

The selection of the different materials for the catoptric behavior characterization, must consider that a high variability exists among the parameters associated to them. Table 1 shows the relation of the materials and their parameters for equation (6). They have been taken from ^{Young & Freedman (2009)}.

In equation (6) it is considered that the energy of the incident wave is completely transferred to the echo and the refraction wave. For that reason, while bigger is the difference between the respective densities of the material and the air, the echo intensity will also be bigger (^{Crawford, 1968a}; ^{Yavorski and Pinski, 1983}; ^{Savéliev, 1984}). Then, the echo intensity in the isotropic source emission focus can be determined according to:

$I_e=R_n{I}_{0exc}{\left(\frac{r_{0exc}\bullet r_{0e}}{r^2}\right)}^2, W/m^2$ (7)

being:

$r_{0e}$ :

Radio of the reflection surface, $m$ .

RESULTS AND DISCUSSION

Behavior of Acoustically Disturbed Medium under the Sensor Excitation

As a result of the transducer excitation, the real shape of the excitation signal is obtained and recorded (see blue curve in Figure 2).

Based on the data set coming from the real signal registration, a fitted mathematical model is obtained (see red curve in Figure 2).

FIGURE 2 Record of sensor excitation signal and its fitted mathematical model. 

The fitted mathematical model of the excitation signal describe it with an adjusted $R^2$ equal to $\mathrm{98,23}\%$ . This model is described analytically by the following function:

$V_{exc}\left(t\right)=\mathrm{4,425}\left[\sin \left(\mathrm{2,5133}\times {10}^5\bullet t+\mathrm{0,4702}\right)+\cdots \right.$

$\left.\cdots +\mathrm{0,2705}\sin \left(\mathrm{7,5399}\times {10}^5\bullet t+\mathrm{0,4702}\right)\right], V$ (8)

According to expression (8) (see the red curve in Figure 2 and the first graph in Figure 3), it is possible to model the behavior of the oscillations in the propagation medium, as well as with the advance of the wave front. So the corresponding wave function is:

${\psi }_{exc}\left(r,t\right)=\frac{\mathrm{1,343}\times {10}^{-9}}{r}\left[sin\left(\mathrm{2,5133}\times {10}^5\bullet t-\mathrm{730,6029}\bullet r+\mathrm{0,4702}\right)+\cdots \right.$

$\left.\cdots +\mathrm{0,2705}\bullet sin\left(\mathrm{7,5399}\times {10}^5\bullet t-\mathrm{2191,8087}\bullet r+\mathrm{0,4702}\right)\right], m$ (9)

FIGURE 3 Result of the mathematical modeling of the behavior of acoustically disturbed medium under the sensor excitation. 

In the second graph, in Figure 3, the behavior of the oscillations in the near proximity of the transducer emission surface is presented. Likewise, in the third graph, in Figure 3, the curve that describes the decrease of the oscillations amplitude, with the advance of the wave front is presented.

In general, in the graphs of Figure 3, a mathematical model running of the acoustic excitation taken place on the propagation medium by the sensor action is presented. In them, the behavior of the wave front is described during the time ( $\mathrm{0,5} ms$ ) that has been programmed for the duration of the medium acoustical disturbance (^{León et al., 2019}).

Influence of Acoustic Reflectivity Coefficient in Echo Formation

In Table 1, the set of acoustic reflectivity coefficients of the selected materials related to the air ( $100 kPa$ , $20°C$ ) is presented Then, no matter the deliberate high variability of the sets of densities and sound speeds, in the set of the acoustic reflectivity coefficients, it presented a very small variation (variation coefficient equal to $\mathrm{0,046}\%$ ).

TABLE 1 Acoustic reflectivity coefficients of several materials in relation to the air ( $100 kPa$ , $20°C$ ) 

Material		Density, $kg/m^3$	Sound speed, $m/s$	Acoustic reflectivity coefficient
Air	( $100 kPa$ , $20°C$ )	$\mathrm{1,2}$	$344$
n
1	Water ( $100 kPa$ , $20°C$ )	$1000$	$1482$	$\mathrm{0,99889}$
2	Aluminum	$2700$	$5092$	$\mathrm{0,99988}$
3	Steel	$7800$	$5064$	$\mathrm{0,99996}$
4	Copper	$8900$	$3516$	$\mathrm{0,99995}$
5	Lead	$11300$	$1190$	$\mathrm{0,99988}$
Variation coefficient, %		$\mathrm{68,32}$	$\mathrm{57,48}$	$\mathrm{0,046}$

Note: The variation coefficient of the densities and the sound speeds shown, do not contemplate the air density and the sound speed in the air.

The obvious interpretation of this result is that they will not have significant differences in the behavior of the echo intensity resultant of the collision of a sound wave (through the air) with one of these materials. In fact, in the worst of the studied cases (water surface at $100 kPa$ and $20°C$ ), the echo will transport the $\mathrm{99,889}\%$ of the incident wave energy, in the zone of the near proximity of the separation limit between air and water.

Echo Intensity Performance

As it has been mentioned previously, the 10CK40T transducer is able to produce a sound pressure level of $120 dB$ for a maximum amplitude of a continuous excitation signal of $20 V$ (peak to peak) and a frequency of $40 kHz$ . Nevertheless, the design parameters of the sensor impose an excitation signal for the transducer of $\mathrm{9,04} V$ (peak to peak), as the one shown in the blue curve of Figure 2. So, this implies that the initial sound intensity of the sensor will be $\mathrm{0,45} W/m^2$ , then it means a sound pressure level in the order of $116 dB$ .

Considering the elements presented previously, it is possible to describe the behavior of the echo intensity with the increase of the distance between the sensor and the separation limit, for the worst of the studied cases (water surface at $100 kPa$ and $20°C$ , with $\mathrm{0,1} m$ of radio). The curves of Figure 4 show this relationship, obtained from equation (7).

FIGURE 4 Echo intensity performance under the sensor design conditions. 

Thus, it can be predicted that at a distance of two meters between the sensor and the separation limit, the echo intensity will be $\mathrm{1,38}\times {10}^{-8}W/m^2$ (echo sound pressure level in the order of $\mathrm{41,4} dB$ ). This guarantees a good sensor performance for a detection zone defined between $\mathrm{0,2}$ and $2 m$ , considering that the sensibility of the 10CK40R transducer is $5\times {10}^{-18}W/m^2$ ( $-63 dB$ ).

CONCLUSIONS

The behavior of the propagation medium response under the acoustic excitation produced by the sensor action was analytically characterized, based on a correlation function fitted to the real excitation signal generated by it, with an adjusted $R^2$ equal to $\mathrm{98,23}\%$ ;

It was determined that the catoptrics conditions of the separation limit between air and a set of materials that differ significantly in their physical properties, do not have significant influence in the resultant echo intensity, because the variation coefficient of the acoustic reflectivity coefficients set obtained is equal to $\mathrm{0,046}\%$ ;

The analytic characterization of sensor response to the echo (considering the worst catoptrics conditions) indicates that the sensor must have a good performance for a detection zone defined between $\mathrm{0,2}$ and $2 m$ , considering that the minimum sound pressure level of the echo is in the order of $\mathrm{41,4} dB$ .

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Received: December 05, 2019; Accepted: September 25, 2020

^*Author for correspondence: Javier A. León-Martínez, e-mail: [e-mail]

Javier A. León-Martínez, Profesor. Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Antihus A. Hernández-Gómez, Profesor Titular, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Ciro E. Iglesias-Coronel, Profesor Titular, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

The authors of this work declare no conflict of interests.

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