1. Introduction
High-voltage cables are the main arteries of urban power supply. Cable accessories connect different sections of cables or cables to other electrical equipment. The stress (compression force) in the cold shrink tube of cable accessories is a key parameter to ensure the stable operation of the power system. Currently, the stress in the cold shrink tube during the installation of cable accessories relies on subjective methods such as visual observation and finger touch, lacking accurate and reliable detection means.
Non-destructive testing methods for residual stress in materials include X-ray diffraction [
1], neutron diffraction [
2], and laser interferometry. Among them, X-ray methods are limited by the measurement platform, making it difficult to perform on-site measurements of parts. Neutron diffraction has low measurement efficiency and requires a reactor or accelerator to provide a neutron source. The measurement equipment is expensive and bulky, remaining in the laboratory stage, unable to achieve large-scale measurements in manufacturing environments. Acoustic detection, as an important non-invasive detection method, has advantages such as good directivity, high resolution, and strong real-time performance, making certain progress in material stress detection.
Critically refracted longitudinal waves (LCR waves) are a type of ultrasonic wave. When ultrasonic longitudinal waves are obliquely incident on solid materials with higher wave speeds, the waves refract. When the incident angle is the first critical refraction angle, the longitudinal waves propagating on the material surface are LCR waves. Compared to other ultrasonic wave modes, LCR wave speeds are more sensitive to material stress changes. The LCR wave measurement method was initially proposed by Srinivasan et al. [
3] and has become the most commonly used ultrasonic method for residual stress measurement [
4,
5,
6,
7,
8,
9]. Since LCR waves propagate along the material surface, they are mainly used for measuring near-surface residual stress in parts. In unidirectional residual stress fields, Jin et al. [
10] measured the sensitivity of LCR wave speeds to stress along different directions. Javadi et al. [
11] experimentally studied the measurement depth of LCR waves using the notch method, showing that the measurement depth decreases with increasing ultrasonic frequency. Wang et al. [
12] proposed an improved ultrasonic method for plane stress measurement using LCR waves and applied a cruciform specimen method combined with a digital image correlation method to confirm the validity of the LCR method. Notably, LCR waves reflect stress changes based on an average value over the acoustic propagation channel, making it difficult to detect stress changes in different regions of the material. Additionally, the application of LCR waves is mainly limited to rigid materials with low acoustic impedance, posing significant limitations in measuring the stress in the cold shrink tube of cable accessories.
When the wavelength of an ultrasonic wave is comparable to the thickness, diameter, and other characteristic dimensions of the part to be measured, the wave undergoes constant reflection, refraction, interference, and mode conversion at the medium boundaries, forming ultrasonic guided waves, such as Lamb waves. Lamb waves exhibit low energy attenuation and long propagation distances, enabling large-area, high-efficiency stress measurement. Safikhanlu et al. [
13] measured the stress in 1050 aluminum samples with a thickness of 0.5 mm, showing that the changes in flight time of S0 and A0 Lamb waves are 3.75 and 1.91 times that of LCR waves, respectively, making them more sensitive to stress measurement. Lim et al. [
14] proposed a dynamic and static load online stress monitoring technology for metal plate structures based on Lamb wave measurement and convolutional neural networks (CNNs). However, Lamb waves exhibit complex dispersion characteristics during propagation, posing numerous problems and challenges in material stress inversion.
Pulse-echo ultrasonic longitudinal wave is an alternative method for stress evaluation. This technology is based on the acoustoelastic effect, where the ultrasonic wave propagation speed is linked to the magnitude of the stresses. Vangi [
15] utilized longitudinal waves in reflection that propagate in a direction perpendicular to the surface of the workpiece and achieved stress measurement in different experimental configurations. The combination of ultrasonic transverse and longitudinal wave testing methods also has new applications in plane stress measurement [
16], but methods based on the attenuation of ultrasonic longitudinal waves for material application measurements are rarely reported.
This paper explores a method for measuring the stress in the cold shrink tube of high-voltage cable accessories based on the attenuation of ultrasonic longitudinal waves. By measuring the transmission and attenuation characteristics of ultrasonic longitudinal waves in multi-layer elastic insulation materials and combining it with theoretical calculations of radial stress in the cold shrink tube, effective measurement of the stress in high-voltage cable accessories can be achieved, providing key technical support for improving the installation process of high-voltage cable accessories.
3. Theoretical Calculation of Multi-Interface Ultrasonic Longitudinal Wave Attenuation
Figure 5 shows the schematic diagram of ultrasonic longitudinal wave interface reflection and refraction. For the cold shrink tube–PVC interface (interface A), according to the theory of reflection and refraction of oblique incident ultrasonic waves, a part of the sound wave will reflect back into the cold shrink tube at a certain angle, while another part will penetrate into the PVC medium. According to Snell’s law of sound wave reflection and refraction [
19],
where
,
, and
are the angles of incidence, reflection, and refraction of the sound wave at the cold shrink tube–PVC interface, respectively.
and
are the sound speeds in the cold shrink tube and PVC medium, respectively.
At the same time, the ratios of the reflected wave sound pressure
to the incident wave sound pressure and the transmitted wave sound pressure
to the incident wave sound pressure at this interface are as follows:
where
and
denote the densities of the cold shrink tube and polyvinyl chloride, respectively.
Similarly, based on Snell’s law and the oblique incidence theory of ultrasonic longitudinal waves, the refraction angles
,
, and
at the liquid coupling agent–cold shrink tube interface, cold shrink tube–liquid coupling agent interface, and cold shrink tube–air interface, respectively, can be obtained. The reflection angle
at the interface D and the ratios of the transmitted wave sound pressure to the incident wave sound pressure
,
, and the ratio of the reflected wave sound pressure to the incident wave sound pressure,
, at the interface D are as follows:
where
and
are the densities of the liquid coupling agent and air, respectively, and
and
are the sound speeds in the liquid coupling agent and air, respectively. The angles
,
, and
are the incident angles of sound waves at the liquid coupling agent–cold shrink tube interface, cold shrink tube–liquid coupling agent interface, and cold shrink tube–air interface, respectively. Meanwhile,
and
as well as
and
represent the transmitted wave sound pressures and incident wave sound pressures at interfaces B and C, and
and
represent the reflected wave sound pressures and incident wave sound pressures at the interface D.
The incident angle at interface B is the angle between the emission probe and the vertical direction . The incident angles , , and at interfaces A, C, and D are all equal to the refraction angle at the liquid coupling agent–cold shrink tube interface.
The number of reflections,
N, of the ultrasonic longitudinal wave in the cold shrink tube changes with the thickness of the cold shrink tube,
, and can be expressed as follows:
During the reflection process, the number of reflections of the ultrasonic longitudinal wave at interface A is N/2 and at interface D is N/2−1. The number of reflections, N, of the ultrasonic longitudinal wave is negatively correlated with the sound pressure at the ultrasonic reception probe, meaning the larger the number of reflections, the smaller the signal amplitude received by the probe.
The length of the propagation path of the ultrasonic wave in the cold shrink tube is . Given that the attenuation coefficient of ultrasound in the cold shrink tube is , the attenuation of the ultrasonic wave along the propagation path can be obtained.
The incident wave sound pressure at interface B,
, is the emission sound pressure
of the ultrasonic probe, and the transmitted wave sound pressure at interface C,
, is the reception sound pressure
of the ultrasonic probe. Therefore, the relationship between them can be summarized as follows:
Because there is a linear relationship between the sound pressure of the ultrasonic probe and the signal amplitude of the ultrasonic probe, the relationship between the emission signal amplitude
of the ultrasonic probe and the reception signal amplitude
can be established as follows:
For the same specifications of cold shrink tube samples in a fully relaxed state, with a cold shrink tube thickness of
, and with the cold shrink tube being empty, the ultrasonic wave during the reflection process only collides with the cold shrink tube–air interface. At this time, the transmitted wave sound pressure at the C interface,
, is the reception sound pressure
, and the relationship between
and
can be expressed as follows:
Therefore, the relationship between the emission signal amplitude
and reception signal amplitude
of the ultrasonic probe under the fully relaxed state of the cold shrink tube can be obtained as follows:
From Equations (14) and (16), the relative amplitude
can be derived as follows:
This relative amplitude can eliminate the interference of reflection attenuation at interfaces B and C and the path attenuation of the ultrasonic wave. The smaller the value of , the greater the attenuation degree of the ultrasonic wave.
According to Equation (17), the theoretical relationship between the relative thickness
of the cold shrink tube and the relative amplitude
can be obtained as follows:
where
is the thickness of the cold shrink tube in the fully relaxed state.
4. Experimental Calibration of Ultrasonic Longitudinal Wave Attenuation in Multiple Interfaces
Figure 6 shows diagrams of cold shrink tubes and PVC standard pieces under four typical conditions. Using cold shrink tubes of the same specification to wrap the PVC standard pieces, the thickness
of the cold shrink tubes decreases as the diameter
of the PVC standard piece increases, as shown in
Figure 6a–c. Additionally, a sample made with a cold shrink tube of the same specification is shown in a completely relaxed state with air inside the tube, as illustrated in
Figure 6d.
The diameter of the PVC standard pieces used in the calibration experiment ranges from 14 mm to 30 mm, with increments of 1 mm, resulting in a total of 17 experimental samples, as shown in
Figure 7. Three sampling positions were randomly selected for each standard piece, and three measurements were performed at each sampling position, resulting in a total of 153 groups of ultrasonic response signals, each containing 500 ultrasonic pulse sequences. At the same time, pulse sequences of the cold shrink tube in a fully relaxed state were measured.
Figure 8 shows the ultrasonic pulse sequences under four typical conditions. It can be seen from
Figure 8a–c that, as the thickness of the cold shrink tube decreases, the amplitude of the ultrasonic pulse gradually decreases, indicating a monotonic relationship between the two. From
Figure 8d, it can be seen that, when the cold shrink tube is empty, its response signal amplitude is significantly greater than that of a tube of the same thickness wrapped with PVC.
The maximum values of the 500 ultrasonic pulse sequences in each group of ultrasonic response signals are extracted and averaged, denoted as . At the same time, the maximum value of the ultrasonic pulse sequence is extracted and averaged when the cold shrink tube is in a relaxed state and empty, denoted as . Thus, the relative amplitude is obtained.
The relative amplitude
measured in the calibration experiment is shown in
Figure 9, where each point corresponds to a group of ultrasonic response signals. By referring to the function model of Equation (19), the relationship between the relative thickness
of the cold shrink tube and the relative amplitude
can be determined as follows:
6. Conclusions
The stress of the cold shrink tube in high-voltage cable accessories is one of the key parameters to ensure the stable operation of the power system. This paper explores a method for measuring the stress of high-voltage cable accessories’ cold shrink tube based on ultrasonic longitudinal wave attenuation. Firstly, a pulse ultrasonic longitudinal wave testing system based on FPGA was designed, where the ultrasonic sensor adopts a single-send, single-receive mode, with an ultrasonic frequency and pulse repetition frequency of 3 MHz and 50 Hz, respectively. The system was optimized to account for the multi-layer structural characteristics of high-voltage cable accessories, ensuring effective wave propagation within the cold shrink tube.
Ultrasonic longitudinal waves attenuate due to reflection and refraction at multi-layer interfaces. This paper divides the attenuation of ultrasonic longitudinal waves into five parts: liquid couplant–cold shrink tube interface attenuation, cold shrink tube–PVC interface attenuation, cold shrink tube–air interface attenuation, cold shrink tube internal propagation attenuation, and cold shrink tube–liquid couplant interface attenuation. The transmission and attenuation characteristics of ultrasonic longitudinal waves in multi-layer elastic media were theoretically studied, and it was found that there is an exponential relationship between ultrasonic longitudinal wave attenuation and the thickness of the cold shrink tube. Through experiments, a measurement model for the thickness of the cold shrink tube was obtained. By establishing a theoretical model of the radial stress of the cold shrink tube and using the thickness of the cold shrink tube as an intermediate variable, an effective measurement of the stress of the cold shrink tube was achieved.
This method not only provides a more accurate and reliable means of stress monitoring in cold shrink tubes but also demonstrates the significant potential of multi-layer elastic media. Future research could focus on expanding its applicability to different materials and structures, such as composite laminate. The advancements could enhance the versatility and effectiveness of ultrasonic longitudinal wave attenuation in stress measurement.