An Improved Calibration Method to Determine the Strain Coefficient for Optical Fibre Sensing Cables

Abstract

The strain coefficient of an optical fibre sensing cable is a critical parameter for a distributed optical fibre sensing system. The conventional tensile load test method tends to underestimate the strain coefficient of sensing cables due to slippage or strain transfer loss at the fixing points during the calibration procedure. By optimizing the conventional tensile load test setup, the true strain of a sensing cable can be determined by using two sets of displacement measuring equipment. Thus, the strain calculation error induced by slippage or strain transfer loss between a micrometre linear stage and sensing cable can be avoided. The performance of the improved calibration method was verified by using three types of sensing cables with different structures. In comparison to the conventional tensile load test method, the strain coefficients obtained by the improved calibration method for sensing cables A, B, and C increase by 1.52%, 2.06%, and 1.86%, respectively. Additionally, the calibration errors for the improved calibration method are discussed. The test results indicate that the improved calibration method has good practicability and enables inexperienced experimenters or facilities with limited equipment to perform precise strain coefficient calibration for optical fibre sensing cables.

1. Introduction

The development of a distributed optical fibre sensing system (DOFSS) provides a new opportunity for geotechnical and structural health monitoring [1]. By integrating a common optical fibre cable into geologic bodies or a structure, the parameters of interest, such as strain or temperature, along the optical fibre route can be obtained [2]. Here, the optical fibre acts as hundreds of thousands of long-distance sensors, which means that there are no monitoring gaps for the monitoring object [3]. Since the 20th century, several types of DOFSS technologies have matured and progressed into commercial tools [4,5,6]. Among them, Brillouin optical time domain analysis (BOTDA) is a widely used technology in civil and geotechnical engineering for strain and temperature measurement due to its high spatial resolution and high measurement accuracy [7,8,9,10,11,12]. To satisfy the engineering requirement, different types of sensing cables have been specially designed and made [13,14,15]. For the case of strain cables, for example, the buffer tubes need to be tightly bonded to the glass fibre to protect the core fibre and ensure that the strain outside the cable is effectively transferred to the fibre core [16,17]. The performance of the cables produced by different manufacturers has a slight difference due to the use of various manufacturing processes or materials. Therefore, calibration is required for engineering to guarantee that the coefficient supplied by the sensing cable manufacturer is accurate and relevant to the interrogator utilized by the optical interrogator.

The strain coefficient is a critical parameter for optical fibre sensing cables [18,19]. This is because the accuracy of the strain coefficient has a great effect on the measurement results, especially for quantitative measurements. Calibration is primarily accomplished by understanding the frequency-dependent transfer function from the environmental strain to the optical fibre strain [20]. The tensile load test [21,22], uniform-strength beam test [23,24], and Yamauchi’s method [25] are the three main approaches used to calibrate the strain coefficient of optical fibres. Among them, the tensile load test method is recommended by the American Society for Testing and Materials because this method is highly flexible and convenient [26]. By fixing the sensing cable at two ends of a stretching table and applying stepwise displacement, the stretched section of the sensing cable yields a uniform strain. The strain coefficient can be determined by fitting a linear relationship between the applied strain and Brillouin frequency shift (BFS) obtained by the optical interrogator. Many laboratories have produced various scales for calibrating devices for different types of sensing cables based on this idea [27,28]. To reduce the error induced by the uncertainty of the length of the sensing cable, Planes et al. [29] changed the strained fibre from a straight line to a folded line by means of several pulleys, resulting in extension of the strained section of the sensing cable up to 8 m in length. In addition, some automated strain calibration devices based on the tensile load test method have been developed [21,27]. However, these improvements do not address the calibration error resulting from the slippage or strain transfer loss of the sensing cable at the fixing point as the fibre is stretched.

The assumption of null strain at the fixing points of the sensing cable in the tensile load test method is not reasonable for accurate strain calibration [30]. Although several fixation methods, including screwing, welding, gluing and other methods, have been explored, slippage at the fixing point between the optical fibre and the displacement table still frequently occurs in the calibration process [27]. Theoretical and experimental studies have proven that strain transfer loss between the sensing cable and the displacement table is unavoidable. The strain transfer loss or slippage leads to a measuring mistake for the change in length of the strained fibre, and, thus, the nominal strain of the sensing cable is overestimated. Therefore, the strain coefficient of the sensing cable is underestimated by using the tensile load test method.

The aim of this article is to provide a simple approach to improve the calibration accuracy for strain coefficient of a sensing cable. An improved strain coefficient calibration device was developed based on the tensile load test method, and three different types of sensing cables were tested. The strain coefficients obtained by the improved calibration method are compared with those obtained by the traditional tensile load test method. Finally, the advantages and limitations of the improved calibration method are discussed.

2. Principle of BOTDA

When light travels through an optical fibre, a small fraction of the light is back scattered due to perturbation, and the scattered light can be divided into three types: Rayleigh scattering, Raman scattering, and Brillouin scattering [31]. BOTDA technology is based on stimulated Brillouin scattering. As shown in Figure 1, one laser beam, called the pump laser, and another laser beam, called the continuous wave probe laser, are injected into an optical fibre core from either end. An acoustic wave will be produced when these two laser beams interact in a specific region of the fibre through a phenomenon called electrostriction. At the same time, the interaction of two light beams leads to a power transfer. The continuous wave probe light will undergo local amplification when the frequency shift between the two light is within the Brillouin gain spectrum of the fibre [32]. The Brillion gain spectrum, as a function of the strain or temperature, can be measured by scanning the frequency of the continuous probe laser. By continuously changing the frequency of the two injected laser beams and analysing the detected backscattering light at the receiving end, the strain or temperature can be expressed as follows:

$$\Delta V_B=V_{B(\epsilon )}-V_{B(0)}=C_{11}\Delta T+C_{12}\Delta \epsilon$$

(1)

where ΔV_B is the BFS difference; V_B(ε) and V_B(0) are the BFS when the strains are ε and 0, respectively; ΔT and Δε are the changes in temperature and strain, respectively; and C₁₁ and C₁₂ are the temperature coefficient and strain coefficient, respectively.

3. Materials and Methods

The improved calibration method described in this article is an enhancement over the tensile load test method. In contrast to the tensile load test setup, the improved calibration method adds a set of strain measuring systems for more accurate measurement of the nominal strain of the strained sensing cable in the calibration process. A schematic diagram of the basic idea for the improved calibration method is shown in Figure 2. Microsliders were bonded with the strained fibre, which can travel along the slide rail when the sensing cable is pulled. Since the movement of the microslider only requires a tiny force, the microslider has nearly no effect on the strain state of the sensing cable. The nominal strain in the optical fibre mid-section can be calculated by dividing the displacement of the two microsliders by the length between the two microsliders. Thus, the strain measurement error induced by slippage or strain transfer loss at the fixing points can be avoided.

3.1. Calibration Setup

Based on the tensile load test method, a device for calibrating the strain coefficient of optical fibres was designed and manufactured. The calibration setup mainly consists of two identical 2500 mm long metal rods, two micrometre linear stages, and two sets of displacement measuring equipment (see Figure 3). The two metal rods are placed side by side, serving as a base, to provide sufficient strength for the tensile load test. The two micrometre linear stages, with an accuracy of 0.02 mm, are fixed at two ends of the metal rods for precise stretching of optical fibres. Because the maximum travel distance of the micrometre linear stage is 25 mm, the two micrometre linear stages can induce an optical fibre elongation of up to 50 mm. Assuming that the elongated section of the sensing cable is 2500 mm, the maximum strain can reach 20,000 με. The two sets of displacement measuring equipment are positioned on the inner side of the two micrometre linear stages. Each set of displacement measuring equipment consists of a laser displacement sensor, an optical fibre holder, and a microslider. As shown in Figure 3a, the optical fibre holder is mounted on the microslider and can freely slide along the slide rail. Thus, the microslider will move in lockstep with the optical fibre elongation during the calibration process. The laser displacement sensor is mounted on the metal rod and used for monitoring the displacement of the optical fibre holder in a non-contact way. The optical fibre holder is tightly attached to the optical fibre and moves along with the elongation of the optical fibre. Since the relative displacement between the optical fibre holder and the fibre can be neglected, the displacement of the optical fibre holder can represent the elongation of the optical fibre in the calibration process. Thus, the truth strain of the optical fibre can be obtained by the laser displacement sensor.

The optical interrogator used in this study is an NBX-6050 produced by the NEUBREX company, Hyogo, Japan, whose operation is based on pulse prepump Brillouin optical time domain analysis (PPP-BOTDA). The strain measurement accuracy is 15 με, as indicated by the manufacturer. According to the user instructions of the device, three main parameters, including fibre length, spatial step, and frequency range, need to be configured to obtain accurate measurement results. The value of the fibre length input to the system will affect the measurement time and analysis algorithm. Normally, the input fibre length needs to be slightly longer than the actual length of the fibre. Considering that the sensing cables used in this study were less than 50 m, the fibre length was set to 50 m. The spatial step, which encompasses the sampling interval and spatial resolution, is closely related to pulse width. In theory, a smaller spatial step allows for the collection of more data points along the optical fibre, resulting in a better measurement accuracy. Therefore, the sampling interval and spatial resolution were set to 5 cm and 10 cm, respectively. The frequency range includes the start frequency (MHz), end frequency (MHz), and frequency step (MHz). In the calibration process, the frequency step was fixed to 5 MHz. Since the central frequency of the optical fibre varies with the strain and temperature, the start frequency (MHz) and end frequency (MHz) need to be adjusted according to the fibre strain condition. The specifications for the instrumentation mentioned above are listed in

Table 1. Specifications or the instrumentation used for the calibration device.

Instrumentation	Specification	Photograph
Optical interrogator	Product type: NBX6050 Measurement: PPP_BOTDA Laser wavelength: 1550 nm Distance range: 50 m, 100 mm … 25 km Measurement frequency range: 9~13 GHz Range of strain measurements: −3%~+4% Strain measurement accuracy: ±15 με/0.75 °C
Metal rod	Material: bearing steel Rail Length: 2500 mm Rail width: 50 mm Weight: 21.18 kg/m Elastic modulus: 200 GPa
Micrometre linear stage	Platform dimension: 125 mm × 125 mm Displacement range: ±12.5 mm Load: 180 N Accuracy: 0.02 mm Minimum scale: 0.01 mm Weight: 1.4 kg
Laser displacement sensor	Model: HG-C1100 Beam diameter: 0.12 mm Measuring centre distance: 100 mm Measurable range: ±35 mm Accuracy: 0.07 mm Dimension (mm): 20 × 44 × 25

.

3.2. Sensing Cables

In general, the bare optical fibre is very fragile and can break easily. To adapt to various situations, the optical fibre is coated with different strengthen or buffer layers. Those layers add strength to the fibre but do not contribute to its optical wave guide properties. However, for a strain optical fibre sensing cable, the cladding may have a slight infect on the measuring result due to the variation thermal expansion parameter of the coating material and strain transfer loss [34]. To verify the practicability of the improved calibration method, three types of optical fibres with different buffering and protection coatings, namely, sensing cable A, sensing cable B, and sensing cable C, were selected for strain coefficient calibration. To verify the practicability of the improved calibration method, three types of optical fibres with different buffering and protection coatings, namely, sensing cable A, sensing cable B, and sensing cable C, were selected for strain coefficient calibration. As given in

Table 2. Specifics for the three types of sensing cables.

Name		Sensing Cable A	Sensing Cable B	Sensing Cable C
Mode		G.652 D	G.652 D	G.652 B
Optical Fibre Diameter		250 μm	0.9 mm	2 mm
Attenuation	1310 nm	0.353 dB/km	0.330 dB/km	--
	1550 nm	0.222 dB/km	0.185 db/km	--
Core-Cladding concentricity error		≤0.6 μm	≤0.6 μm	--
Manufacturer		HengTong group, Jiangsu, China	Yangtze Optical Fibre and Cable Joint Stock Limited Company, Wuhan, China	NanZee Sensing company, Nanjing, China
Structure
Photograph

, sensing cable A is a telecommunication grade, G.652. D single-mode optical fibre cable with a diameter of 250 μm produced by HengTong group, China. The sensor is usually only utilized in laboratory studies due to its fragility. Sensing cable B is a G.652. D single mode fibre with a tight polymer buffer and a diameter of 0.9 mm, which is produced by the Yangtze optical fibre and cable joint stock limited company. Because sensing cable B is thin and flexible, it can be easily attached to the objects being monitored. Sensing cable C is a G.652. B single-mode optical fibre cable with a polyurethane jacket as a buffer layer and a diameter of 2 mm. Sensing cable C, manufactured by NanZee Sensing Company, Nanjing, China, is specially designed for physical model tests.

Considering that each of the three sensing cables has a low tensile strength, the adhesive method was utilized to fix the sensing cables at the ends of the calibration setup. To save time, the sensing cables were first adhered to acrylic plates using structural acrylic adhesive and then cured for 24 h, following which the acrylic plates were mounted onto the micrometre linear stages through bolts.

3.3. Calibration Process

The test was conducted in a laboratory where the room temperature was maintained constant at 25 °C by an air conditioner. A digital hygrometer was used to monitor the environmental conditions in real time. Following the suggestion of standard ASTM F3079–4 [26], the initial condition for the sensing cables shall entail an elongated state. After installing sensing cables on the calibration device, the micrometre linear stage adjuster was scrolled to apply a tension force to the sensing cables. The initial length of the strained optical fibre was measured by using a short steel tape measure. The distance between the two micrometre linear stages is defined as L_0, and the distance between the two micro sliders is defined as L_m0. Then, the sensing cable was stretched by approximately 2~4 mm in each step, resulting in a strain increase of approximately 1000~2000 με. For each elongation step, the displacements for the two micrometre linear stages and the variation in the value indicated by the two laser displacement sensors was recorded manually. Meanwhile, the BOTDA interrogator NBX-6050 was utilized to determine the BFS for the sensing cable at different strain conditions. The nominal strain of the sensing cable measured by the tensile load test method and the improved calibration method were determined by using Equations (2) and (3), respectively:

$${\epsilon }_i=\Delta L_i/L_0$$

(2)

$${\epsilon }_{m,i}=\Delta L_{mi}/L_{m0}$$

(3)

where ΔL_i is the displacement of the micrometre linear stages, ΔL_mi is the elongation of the mid-section measured by the two laser displacement sensors, ε_i is the strain calculated by the tensile load test method, and ε_mi is the strain calculated by the improved strain calibration method. The strain coefficient for each sensing cable was determined by a linear regression between BFS and the nominal strain. Furthermore, the calibration error was subjected to a thorough examination.

4. Results and Discussion

The three sensing cables were calibrated both with the tensile load test method and the improved calibration method. In addition, the performance of the BOTDA interrogator and the sensing cables was tested prior to calibration. During the calibration process, the room temperature was kept at 25.5 °C~25.7 °C with a humidity of 61%. Thus, the effect of temperature on the calibration results can be ignored. For sensing cable A, the tensile load was applied by using only one micrometre linear stage. The tensile loads for sensing cables B and C were applied by simultaneously operating the two micrometre linear stages at the ends of the calibration setup. As the elongation increases, sensing cable A loses its signal when the maximum strain reaches 10,655.66 με. Visual inspection revealed that the acrylate layer peeled off from the core of the fibre at the fixing point (see Figure 4). This phenomenal is mainly induced by strain transfer loss. With the displacement of the micrometre linear stage increase, the strain difference between the acrylate layer and the core of the fibre increases at the fixation point. When the strain of the acrylate layer excessed its maximum allowable strain, the acrylate layer will peel off from the core of the fibre. Sensing cable B and sensing cable C still work normally when the strain reaches 20,296.4 με and 19,266.8 με, respectively.

4.1. Quality Control

4.1.1. BOTDA Interrogator

The strain measurement is subject to a measurement repeatability error, which is highly related to the measurement accuracy of the BOTDA interrogator. Normally, accuracy is referred to the closeness of the measurements to a specific value [35]. In this paper, the performance of the BOTDA interrogator is defined as the statistical variance in a repeat measurement of the BFS along the optical fibre. The BFS readings were taken by using the same interrogator, configuration, and operator in a very short time. All three sensing cables were measured five times at a free strain state. The spatial variation of the BFS along the optical fibre was obtained by calculating the standard deviation for the multiple readings. As shown in Figure 5, the standard deviations in BFS are always below 0.75 MHz for sensing cables A and C at the mid-section. The standard deviation in BFS for sensing cable B spans a wide range from 0 MHz to 1.65 MHz. This may be related to the fact that sensing cable B exists in a ring-wound state during the test. The strain measurement accuracy provided by the equipment manufacturer is ±15 με (equal to approximately ±0.75 MHz). Therefore, it can be determined that the strain measurement accuracy of the BOTDA interrogator is consistent with the factors reported by the manufacturer.

To further investigate the effect of the strain measurement accuracy for small strain measurements, the first reading of the BFS was treated as a baseline, and the other four readings were treated as observed values. The BFS difference (ΔV_B) is defined as the difference between the baseline and observed values. To facilitate understanding, the difference in the BFS was converted to a strain with a strain coefficient of 0.05 MHz/με. Figure 6 shows the results obtained. The strain errors mainly ranged within −20~+30 με, −40~+30 με, and −20~+20 με for sensing cables A, B, and C, respectively. This means that the effect of the strain measurement error needs to be considered when measuring strain values that are less than the strain errors. Only when the measured strain is much larger than the strain errors can the strain measurement error be ignored.

4.1.2. Uneven Strain of the Sensing Cables

Figure 7 illustrates the distribution of the BFS for the three sensing cables at different strain conditions. When the strained section of the sensing cable undertakes a small stress, the BFS along the optical fibre changes slowly from the strain-free section to the strain section. As the stretch length increases, the BFS changes sharply from the strain-free section to the strain section. When the stretching exceeds a certain distance, for example, sensing cable A, the strained section undergoes an overall shift to the right by 5 cm (see Figure 7a). This is due to the change in length produced by the stretching of the sensing cable.

Ideally, the BFS should remain constant within the elongated section of the sensing cable. However, in practice, unexpected BFS fluctuations are often observed, and the influence on the strain coefficient calibration is not assessed. The standard deviations for the BFS of the sensing cables are shown in Figure 8a. With increasing strain, the standard deviations for the BFS for all three sensing cables slightly increase. Especially for sensing cable B, the standard deviation changes from 1.089 MHz to 2.486 MHz. Sensing cable A has the lowest standard deviation between 0.569 MHz and 1.653 MHz, and sensing cable C has the highest standard deviation between 2.543 MHz and 3.431 MHz. These values are larger than the BOTDA interrogator-induced errors. Therefore, the BFS fluctuation may be related to nonuniform bonding of the coatings or buffering.

In engineering applications, the incremental BFS profile obtained from the original BFS profile is commonly used to assess the performance of the structure. Hence, the BFS difference is a good indicator to illustrate the effect of a sensing cable’s uneven strain on the calibration results. Figure 8b shows the standard deviation of the BFS difference for sensing cables A, B, and C at different strain states. With increasing strain, the standard deviations for the BFS difference for all three sensing cables slightly increase. The standard deviations for the BFS difference at the largest strain states are 1.197 MHz, 1.705 MHz, and 1.70 MHz for sensing cables A, B, and C, respectively. This is comparable to the systematic error of the BOTDA interrogator. Therefore, the three sensing cables are considered to have acceptable uneven strains, and the measurement error for V_B(ε) can be ignored.

4.2. Error Analysis

In the strain calibration process, two main factors that affect the calibration accuracy include the measurement error for the sensing cable’s original length and the measurement error for the change in length of the strained fibre. Since the error magnitude is related to the measurement range of the instrumentation, the relative uncertainty was utilized to highlight the effect of the two factors on the calibration results.

4.2.1. Original Length-Induced Error

The measurement accuracy of the original length of the calibration section L₀ is subject to the accuracy of the measuring tools used and the operating error. In this study, the length of the sensing cables was manually measured by using a short steel tape measure with an accuracy that met Japanese industrial requirements Class I. According to the data given in the instruction manual, the maximum error is expected to be ±(0.2 + 0.1 L) mm, where L is the measuring length. Assuming that the true length of the fibre is 2 m, the error induced by the steel tape measure is ±0.4 mm. The magnitude of human-induced error is limited by the skill of the operator. Considering that the lowest count of the steel tape measure is 1 mm, we assume that the human-induced error is ±2 mm. Thus, the total measurement error for the original length of the calibration section L₀ is ±2.4 mm. The strain error induced by the measurement error for L₀ can be calculated using

$${\epsilon }_e=\frac{\Delta l}{L_0}-\frac{\Delta l}{L_0^{\prime }}$$

(4)

where ε_e is the strain error induced by the measurement uncertainty of the original length of the sensing cables, L₀ is the true length of the sensing cable, L^′₀ is the measured length of the sensing cable, and Δl is the elongation length of the sensing cable. When Δl is 50 mm, it will lead to a deviation of ±30.04 με in the obtained strain. The relative uncertainty induced by the error in the measurement of the original length can be obtained using the following equation:

$$U_1=\frac{{\epsilon }_e\times C_{12}\times L_0}{\Delta l}$$

(5)

where U₁ is the relative uncertainty of the strain coefficient induced by the uncertainty of the fibre original length and C₁₂ is set to 0.05 MHz/με. Therefore, the U₁ is approximately ±6.008 × 10⁻⁵ MHz/με.

4.2.2. Slippage or Strain Transfer Loss Induced Error

The primary objective of the improved calibration method proposed in this article is to eliminate slippage- or strain transfer loss-induced error in the calibration process. By using two sets of displacement measuring equipment, the strain error induced by the measurement error for the micrometre linear stages is avoided. Figure 9 illustrates the strain difference along with the strain increase for the calibration process. Here, the strain difference is defined as the difference between the strain calculated by the tensile load test method and the improved calibration method. For sensing cable A and sensing cable B, the strain difference increases almost linearly with fibre elongation. For sensing cable C, the strain difference first increases linearly, followed by an exponential growth trend. This indicates that the cable jacket of sensing cable C undergoes nonlinear elastic deformation when the strain exceeds approximately 15,000 με. Although no significant fibre deformation was observed at the fixing point, the monitoring data indicated that strain transfer losses did occur. The magnitude of the strain transfer loss for the three sensing cables is mainly related to the material properties of the sensing cable jacket and the fixing method.

Additionally, the occurrence of slippage is very hard to predict. Therefore, the improved calibration method proposed in this article is necessary and significant. The relative uncertainty induced by the measurement error for the change in length of the strained fibre can be obtained from

$$U_2=\frac{{\epsilon }_d\times C_{12}\times L_0}{\Delta l}$$

(6)

where U₂ is the relative uncertainty of the strain coefficient induced by the uncertainty of the change in length of the strained fibre, ε_d is the strain difference, and C₁₂ is set to 0.05 MHz/με. Since there is a positive correlation between the strain difference and BFS, U₂ is equal to the slope in Figure 5. Therefore, the relative uncertainties for the strain coefficient induced by strain transfer loss for sensing cables A, B, and C are 0.3231 με/MHz, 0.3808 με/MHz, and 0.3505 με/MHz, respectively. The U₂ values for sensing cables A, B, and C can be converted into units of MHz/με by using a coefficient of 0.05 MHz/με to give 8.074 × 10⁻⁴ MHz/με, 9.52 × 10⁻⁴ MHz/με, and 8.763 × 10⁻⁴ MHz/με, respectively.

4.3. Calibration Results

The strain coefficients obtained for the three sensing cables by using the improved calibration method and the tensile load test method are illustrated in Figure 10. The red and blue lines represent the results obtained from simple linear regression for the improved calibration method and the tensile load test method, respectively. For both techniques, the goodness-of-fit (R²) for linear regression between strain and BFS was more than 0.999. All the sensing cables’ strain coefficients are close to 0.05 MHz/με, which is consistent with the results reported by most researchers. However, the strain coefficients obtained from the improved calibration method are always larger than those obtained from the tensile load test method for the three sensing cables. The difference in the strain coefficient obtained using the two different methods is 7.8 × 10⁻⁴ MHz/με, 1.04 × 10⁻³ MHz/με, and 9.5 × 10⁻⁴ MHz/με for sensing cables A, B, and C, respectively, which is consistent with the slippage- or strain transfer loss-induced error. Given that the original length of the sensing cable leads to an inaccuracy of approximately ±6.008 × 10⁻⁵ MHz/με, the final strain coefficients for the three sensing cables should be (5118 ± 6.008) × 10⁻⁵ MHz/με, (5060 ± 6.008) × 10⁻⁵ MHz/με, and (5112 ± 6.008) × 10⁻⁵ MHz/με, respectively. Compared with the tensile load test method, the strain coefficients obtained using the improved calibration method show relative improvements of 1.52%, 2.06%, and 1.86% for sensing cables A, B, and C, respectively.

4.4. The Advantages and Disadvantages of the Improved Calibration Method

In contrast to the tensile load test method, the improved calibration method does not assume null strain at the ends of the bonding length. In fact, slippage or strain transfer loss at the end of the fixation section is difficult to eliminate in the calibration process [3]. An imperceptible error can lead to an underestimation of the strain coefficient, especially for novices. In our preliminary test, significant slippage occurred between the sensing cable and the cured adhesive (see Figure 11). The obtained strain coefficient was 0.0419 MHz/με, which is much smaller than the true value. In this paper, the difference in the strain coefficients obtained by the improved calibration method and the tensile load test method is no more than 2%. This is mainly due to the use of a suitable glue and sufficiently large calibration setup in the experiment. The error induced by the uncertainly in the initial length was greatly reduced. Furthermore, the sensing cables used in this study are very thin, which contributes to a reduction in error due to strain transfer loss at the fixing points. In short, the improved strain calibration method proposed in this study can be used to solve the above problems and make the calibration results more accurate by only slightly improving the tensile load test setup.

Due to the limited pulling force available from the micrometre linear stage, the calibration setup is only suitable for thin sensing cables with low elastic models. It is necessary to develop pulling devices with a high pulling force for sensing cables with strength members. Additionally, the use of glue to secure sensing cables requires a long curing time and it is only capable of withstanding a small pulling force. Therefore, it is vital to build an easy-to-use sensing cable clamp that can provide adequate friction while avoiding damage to the fibre. Finally, developing fully automated calibrating equipment is a significant area of research. Not only can the calibration efficiency be improved, but human operation errors can also be eliminated.

5. Conclusions

The problem of underestimating strain coefficients for sensing cables in the tensile load test method was studied. According to our empirical and theoretical analysis, the calibration error is primarily related to the slippage or strain transfer loss between the sensing cable and the micrometre linear stage during the calibration procedure. To address this issue, this paper proposes an improved calibration method based on the tensile load test method to increase the accuracy of the strain coefficient calibration results. Based on the test results, the following conclusions can be drawn:

• An improved strain coefficient calibration device can be developed by adding two sets of displacement measuring equipment to the traditional tensile load test setup. Thus, the strain in the mid-section of the strained sensing cable can be obtained more accurately.
• Although no slippage is observed at the fixing point in the calibration process, the results from error analysis indicate that the source of the strain coefficient calibration error is mainly due to inaccurate measurements of the displacement by the micrometre linear stage. Therefore, it can be presumed that strain transfer loss occurs between the sensing cable and micrometre linear stage.
• The performance of the improved calibration method was verified by using three types of optical fibre sensing cables. In comparison to the traditional tensile load test method, the strain coefficients obtained for sensing cables A, B, and C by using the improved calibration method are improved by 1.52%, 2.06%, and 1.86%, respectively. Although the improved calibration method only shows a slight improvement in the strain coefficient calibration results compared to the conventional tensile load test method used in this test, the improved calibration method is significant because it eliminates potential calibration errors. This enables inexperienced experimenters or facilities with limited equipment to precisely calibrate the strain coefficient of a sensing cable.