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).
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
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:
being:
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Amplitude of the main harmonic component of
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Non-dimensional relationship among the amplitudes of the main and secondary harmonic components respectively |
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Phase of
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Nominal frequency of the transducer,
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Time,
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The parameters
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:
where:
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Wave propagation distance in the instant of time
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Radio of the transducer emission surface,
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The wave number of the main harmonic component in the wave function (2), is determined according to:
Thus, the oscillations amplitude on the emission surface of the transducer will be:
Considering the sensor design conditions, the wave intensity on the emission surface of the transducer can be determined according to:
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:
where:
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Numeric identification associated to the corresponding material (see Table 1); |
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Density of the corresponding material,
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Sound speed in the corresponding material,
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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:
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).
The fitted mathematical model of the excitation signal describe it with an adjusted
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:
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 (
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 (
Material | Density, |
Sound speed, |
Acoustic reflectivity coefficient | |
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Air | ( |
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1 | Water ( |
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2 | Aluminum |
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3 | Steel |
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4 | Copper |
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5 | Lead |
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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
Echo Intensity Performance
As it has been mentioned previously, the 10CK40T transducer is able to produce a sound pressure level of
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
Thus, it can be predicted that at a distance of two meters between the sensor and the separation limit, the echo intensity will be
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
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
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