Among the optical fiber sensors, fiber Bragg grating (FBG) has found remarkable attraction and many applications due to its ability to measure all environmental parameters, high accuracy and sensitivity, easy installation, and low price. The sensor's ability to simultaneous measurement of two environmental (physical) parameters, especially strain and temperature, has made it even more demanding. Recently, two techniques based on FWHM and peak power changes of FBG spectrum have been proposed for simultaneous measurement of temperature and strain that use a uniform FBG on a tilted cantilever beam. In this article, we evaluate and compare both techniques and demonstrate that both are equivalent and have almost similar behaviour but do not have the same applications. We find that at the low strain range the peak power technique is more accurate while at the high strain range the FWHM technique has better accuracy. Therefore, in applications such as monitoring civil structures that have a high strain range, it is better to use the FWHM technique and in some delicate industrial applications with a low strain range, the peak power technique is used.
Research Article
Comparison of FWHM and peak power of FBG spectrum for simultaneous measurement of strain and temperature based on tilted cantilever beam
https://doi.org/10.21203/rs.3.rs-1622423/v1
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Fiber Bragg grating
FBG sensor
simultaneous measurement
cantilever beam
comparison
The root of optical fiber sensor technology is directly associated with the development of the modern lasers attained in the 60s. Six decades have passed since the beginning of research on fiber sensors and today these sensors are used in various civil and industrial fields [1–7]. These sensors have several potential benefits that make them attractive for a variety of sensing applications. They are typically small in size, immune to electromagnetic interference, passive, resistant to harsh environments, and can perform distributed sensing [8–11]. Among optical fiber sensors, fiber Bragg grating (FBG) has found many applications due to its ability to measure all environmental parameters such as temperature, strain, stress, bending, humidity, etc. Also, easy installation, cost-effectiveness, and high accuracy and sensitivity of the FBG sensor are other reasons for the acceptance of these sensors [12–13]. However, some limitations in uniform FBG has led to a decrease in performance of FBG sensors that most important of them is limitation in the simultaneous measurement of two physical parameters, especially temperature and strain, by a uniform FBG [14–16]. Because by changing both temperature and strain parameters, only the wavelength in the FBG reflectance spectrum changes, so independent measurement of these two parameters is necessary to achieve high accuracy [17–18].
In the last two decades, many techniques have been proposed for the simultaneous measurement of strain and temperature with FBG sensors. The proposed techniques fall into three categories. In the first category, more than one uniform FBG is used [19–21] and in the second category, only one special (non-uniform) FBG is adopted for simultaneous measurement [22–26]. Both categories have disadvantages such as the use of more than one uniform FBG, complicated setup and fabrication of special FBGs, excessive use of spectral sources, and difficult accessibility. In the recently introduced and well noticed third category, only a uniform FBG is used with the help of a mechanical trick for the simultaneous measurement of temperature and strain [27–29].
In this new category, unlike the previous two categories, there are not many methods and only a few limited techniques have been reported such as a half of the FBG has been tied to a host structure, while the other half of the FBG has been left free. In this technique, the temperature can be calculated by the Bragg wavelength shift of unbounded FBG, while the strain is determined by the Bragg wavelength shift of the bonded FBG. As explained in the work, measurements have errors since it is assumed that the unbounded FBG does not experience any strain while in reality, it experiences the strain [28–29]. The next technique uses the reflection spectrum power of the side-lobes for simultaneous measurement of strain and temperature. This method requires very accurate spectrometers to be able to correctly detect the peak power of the side-lobes [27].
In the last two reports published by our group, which also fall into the third category, a single (uniform) FBG pasted on a tilted cantilever beam was used to simultaneously measure strain and temperature on a sample under test [30–31].The cantilever beam which is placed on the sample structure, causes the uniform strain applied to the structure to become a non-uniform strain on the FBG. Non-uniform strain along with the uniform FBG also results in a change in FWHM and peak power of the FBG reflection spectrum. Therefore, the cantilever beam has caused the two indicators of FWHM physics and reflective peak power to react to strain changes. In these two reports, temperature changes are obtained by displacement of Bragg wavelengths, and strain changes can be calculated using both the FWHM technique and the peak power technique. In this paper, we will compare the sensitivity of two indicators of FWHM and peak power change to strain in the tilted cantilever beam technique and illustrate which indicator is a more accurate and sensitive indicator for the simultaneous measurement of temperature and strain.
Cantilever beams are a rigid tool and mechanical plat-like structure that anchors at one end to support (usually vertical) that protrudes from it. Therefore, the important point about cantilever beams is that the strain distribution along them is linear and non-uniform [32–33]. In this case, the cantilever beam is attached and fixed with a slight angle θ on a sample under test and a uniform FBG is attached on this tilted cantilever beam. The strain applied to the sample structure (εs) and the strain experienced at each part of the cantilever beam (εc) has a significant relation with each other. This relation depends on the angle θ, length, thickness and the inherent properties of the cantilever beam. In addition, the M coefficient is an experimental strain to show these dependencies that can vary from 0 to 1.
The strain distribution in the cantilever beam is such that its end experiences the maximum strain equal to M.εs.cosθ and this strain decreases linearly until it reaches 0 at the free end. Therefore, according to the location of FBG on the cantilever beam, it can be seen how much of the uniform strain applied to the sample enters the beginning and end of FBG.
|
Strain in sample εs (µε) |
Maximum Strain in fixed-end of cantilever (µε) M.εs.cosθ |
Maximum wavelength shift, 1550nm |
Maximum FWHM, 0.39nm |
Maximum ∆FWHM (nm) |
Maximum peak power, P |
Maximum ∆P |
|---|---|---|---|---|---|---|
|
100 |
86.6 |
0.0453 |
0.3905438 |
0.0005438 |
0.998525 |
0.001475 |
|
200 |
173.2 |
0.0906 |
0.392170586 |
0.002170586 |
0.994475 |
0.005525 |
|
300 |
259.8 |
0.136 |
0.394896 |
0.004896 |
0.98762 |
0.01238 |
|
400 |
346.4 |
0.1812 |
0.39874 |
0.00874 |
0.97823 |
0.02177 |
|
500 |
433 |
0.2265 |
0.4037254 |
0.0137254 |
0.966344 |
0.033656 |
|
600 |
519.6 |
0.272 |
0.40988425 |
0.01988425 |
0.95221 |
0.04779 |
|
700 |
606.2 |
0.3171 |
0.4172314 |
0.0272314 |
0.935953 |
0.064047 |
|
800 |
692.8 |
0.3624 |
0.425854 |
0.035854 |
0.917854 |
0.082146 |
|
900 |
779.4 |
0.4077 |
0.4357478 |
0.0457478 |
0.8981 |
0.1019 |
|
1000 |
866 |
0.453 |
0.4469658 |
0.0569658 |
0.876965 |
0.123035 |
For example, if the length of the tilted cantilever beam is 6cm and the length of FBG is 2cm, and the FBG is in the middle third of the cantilever beam, a strain equal to (2/3).(M.εs.cosθ) is felt at the beginning of the FBG and (1/3).(M.εs.cosθ) is achieved at the end of the FBG, which show the maximum and minimum strain reached to FBG, respectively. As a result, the FWHM and peak power will change with the strain applied to the sample under test according to Table 1. In this article, we assumed that θ = 30 and M = 1 (it is assumed that all applied strain to the sample is transferred on the fixed end of the tilted cantilever beam).
According to Table 1, the relationship between the maximum Bragg wavelength shift and the strain applied to the sample is linear and is equal to 0.000453 nm/µε, while according to Fig. 1, the relationship between the maximum ΔFWHM / power peak and the strain applied to the sample is nonlinear as shown in Eq. (1):
By knowing the maximum FWHM and peak power variations, the strain applied to the sample is obtainable as can be seen in Eq. (1). Of course, it should be noted that Eq. (1) is obtained according to our conditions (location on the cantilever beam, FBG and cantilever length) and with the change of each condition, the coefficients of the Eq. (1) also change. The Eq. (1) would have temperature dependency if there was a non-uniformity in the temperature distribution in the FBG in addition to strain distribution.
Whenever a uniform strain is applied to a structure, in addition to changing the Bragg wavelength (0.453 pm/µε), it leads to changes in FWHM / power peak (nonlinear), but a temperature change in the same structure only leads to a change in Bragg wavelength (0.0142 nm). At a place where both strain and temperature change, by obtaining the maximum change in wavelength and FWHM / peak power of the FBG reflection spectrum, the temperature and strain applied to the sample under test can be obtained by two equations with two unknowns according to Eq. (2) as:
Figure 2 shows the unknown environmental conditions for our sample and we intend to simultaneously calculate the temperature and strain changes applied to the sample. In this figure, the maximum wavelength shift is 0.646nm and the maximum ∆FWHM and ∆P are equal to 0.03448nm and 0.08215. According to Eq. (5), if we consider ∆FWHM and ∆λ, the strain and temperature variations of the sample under test are obtained equal to 785µε and 20.45℃, while if consider ∆P and ∆λ, the strain and temperature variations are 788µε and 20.35℃ respectively.
As we saw from the example above, the results of these two techniques are slightly different from each other. Now we want to use simulation to find out which technique is more accurate.
|
Strain in sample εs (µε) |
εs, FWHM (µε) |
εs, Peak Power (µε) |
|---|---|---|
|
100 |
94 |
100 |
|
200 |
198 |
201 |
|
300 |
296 |
312 |
|
400 |
393 |
425 |
|
500 |
491 |
536 |
|
600 |
590 |
647 |
|
700 |
688 |
755 |
|
800 |
787 |
860 |
|
900 |
888 |
963 |
|
1000 |
989 |
1063 |
As shown in Table 2, both two techniques are equivalent and have almost similar behaviour. The peak power technique gives more accurate results in the low strains range and the FWHM technique is more accurate for the high strains range. By plotting the strain difference between these two techniques relative to the original strain (Fig. 3), it is possible to determine what range of strains each of these two techniques is suitable for measuring.
According to Fig. 3, it can be seen that for strains less than 200 µε, the peak power technique shows more accurate results, and with increasing strain applied to the sample, it is better to use the FWHM technique to measure temperature and strain simultaneously. Of course, it should also be noted that from an empirical perspective, measuring the maximum peak power variation is much easier than measuring maximum ∆FWHM, because to measure the maximum ∆P, unlike ∆FWHM, you do not need complex tools.
In the tilted cantilever beam method, temperature and strain can be obtained simultaneously in two techniques, FWHM and peak power. Although the two techniques behave almost identically, the sensitivity and accuracy of the measurements are different. The FWHM technique is suitable for strains in the range of 200µε and below, and the peak power technique is best used for strains above 200µε. Of course, the facilities and tools needed to measure strain and temperature simultaneously with the peak power technique are much more accessible than the FWHM technique.
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