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A fiber grating sensor capable of distinguishing between temperature and strain, using a reference and a dual-wavelength fiber Bragg grating, is presented. Error analysis and measurement uncertainty for this sensor are studied theoretically and experimentally. The measured root mean squared errors for temperature T and strain ε were estimated to be 0.13 °C and 6 με, respectively. The maximum errors for temperature and strain were calculated as 0.00155 T + 2.90 × 10^{−6} ε and 3.59 × 10^{−5} ε + 0.01887 T, respectively. Using the estimation of expanded uncertainty at 95% confidence level with a coverage factor of

The advantages of fiber optic sensors include light weight, small size, immunity to electromagnetic interference (EMI), large bandwidth, environmental ruggedness and electrical and optical multiplexing capability. Thus, fiber optic sensors are ideal for the applications in potential smart structures and materials. Fiber Bragg gratings (FBGs) have many wide applications, ranging from tele-communications to optical fiber sensors. Though Bragg grating sensors (BGSs) have offered a variety of potential advantages over their conventional counterparts, their widespread practical use has been plagued by their inability to effectively discriminate between temperature and strain fields. A number of attempts to overcome this limitation have been demonstrated [_{s}_{s}^{−1}, where

Though BGSs have offered potentially numerous advantages over their conventional electrical and mechanical counterparts, their widespread use has been limited by their inability to discriminate effectively between temperature and strain fields, and this poses serious problems for sensors system designed to monitor quasi-strain signals, as temperature variations along the fiber link will induce indistinguishable thermal-apparent strain signals. It is apparent that measurement of one wavelength shift from a single grating will not determine those two variables simultaneously. A number of methods that separate temperature- and strain-induced wavelength shift and overcome this limitation have been proposed and demonstrated, including the use of reference grating [_{1T},κ_{2T}) and strain (κ_{1ε},κ_{2ε}) at the same location on the structure are different. Dual-wavelength technique requires two broadband sources to address each sensor and suitable wavelength demodulation system (WDS) at the output. The change in the Bragg center wavelengths Δλ_{i} of the two gratings from the changes in temperature (ΔT_{i}) and strain (Δε_{i}) is given by the following matrix expression
_{i}_{ε} = _{i}_{iT}_{i}

We have developed a simple and low-cost optical fiber sensor for this purpose [_{1} and λ_{2},) and a packaged reference grating (λ_{3}). The bare grating pair was constructed by fusion splicing two fiber Bragg gratings in cascade with different Bragg wavelengths. The spliced portion of the grating pair was glued into a quartz tube in order to prevent the relatively brittle spliced or fused portion from being damaged or broken. To protect the reference grating from mechanical deformation and damage, a method of packaging the bare fiber Bragg grating with a stainless steel tube was applied. In the packaging process, the reference grating was first bonded to a quartz substrate with an adhesive and the substrate with the reference grating was inserted into a stainless steel tube, and the both ends of the stainless tube was then glued and sealed with elastic epoxy glue. In this sensor structure, the free end of the fiber was secured with adhesive tape to avoid any unwanted movement or twisting. The three fiber Bragg gratings at wavelengths of λ_{1}, λ_{2}, λ_{3} were interrogated using a broadband ASE source and an optical spectrum analyzer (OSA). A fiber coupler was used for coupling the reflected light signals of the sensor to the OSA. The reference grating was used to measure only the temperature effect. The shift in Bragg wavelength λ_{3} from temperature changes is given by
_{3} can be used to determine uniquely the local temperature provided that the temperature coefficient κ_{3T} is well known. The grating pair was fabricated by splicing two fiber Bragg gratings with wavelengths, λ_{1} and λ_{2}, respectively. The wavelength shifts Δλ_{i} from temperature (ΔT_{i}) and strain (Δε_{i}) changes were calculated using

It can be seen that the errors measured in temperature and strain are determined primarily by the resolution effect of optical spectrum analyzer and the errors in estimation of temperature and strain coefficients (see _{1ε} = 0.914 pm/με; κ_{2ε} = 0.918 pm/με; κ_{1T} = 10.4 pm/°C; κ_{2T} = 12.1 pm/°C). Although the use of a fast and high resolution grating interrogation system is feasible, to build such a sensor system is costly. A simple and cost-effective method for improving the performance is to use the reference grating as an independent temperature sensor. The reference grating can be used to reduce unnecessary errors induced from the grating pair and to improve the accuracy of the temperature measurement.

According to the error analysis technique presented by Jin _{1} and λ_{2}. The maximum errors in temperature T and strain ε is formulated as:
_{1T}κ_{2ε} − κ_{2T}κ_{1ε}. The maximum measurement errors of δT and δε were calculated as 0.13 °C and 1.6 με, respectively. For the second case, assuming that the measurement errors in λ_{1} and λ_{2} may be neglected (δλ_{1} = δλ_{2} = 0) and the maximum errors are in all the coefficients, the maximum relative errors for δT/T and δε/ε is expressed as:

Thus, the maximum relative errors for δT/T and δε/ε were calculated as 0.0016 + 2.90 × 10^{−6} ε/T, and 3.59 × 10^{−5} + 0.0188 T/ε, respectively. In

Secondly, considering all the measurement errors involved in determining the coefficients and the precision of wavelength measurement, the maximum relative errors for δT/T and δε/ε is given by
^{−4} ε/T, and 1.75 × 10^{−3} + 0.021 T/ε, respectively. As shown in

However, in Reference [_{max} or δε_{max}, to be relative maximum error (δT/T or δε/ε, unit less). The value of relative maximum error (percentage) is not always between 0 and 100. It is not recommended to use the relative maximum error when the temperature or strain is zero since relative maximum error becomes singular in this situation. Actually, for our laboratory testing data, the controlled temperature and strain ranges were 25∼113 °C and 100∼1,600 με, respectively. There was no singular problem for our testing results. Therefore, using maximum errors instead of relative maximum errors could be a better way to characterize measured quantity. The maximum errors for temperature and strain are expressed by

With _{max} and |δε|_{max}, were estimated as 0.00155 T + 2.90 × 10^{−6} ε and 3.59 × 10^{−5} ε+ 0.01887 T, respectively.

The measurement uncertainty for temperature and strain simultaneous measurements using dual wavelength grating method was studied as estimation of standard uncertainty, combined standard uncertainty, and expanded uncertainty [

The models for temperature and strain difference were shown in

Since the measurement resolution of wavelength shift using the ANDO AQ6331 OSA was ±0.05 nm, the uncertainty on the FBGs measurement was as large as 3 pm [_{Δλ1} = infinity, and ν_{Δλ2} = infinity, therefore;

Standard uncertainty, u_{Δλ1}

Standard uncertainty, u_{Δλ2} = 3 pm/2 = 1.5 pm;

Based on _{1} and λ_{2}, respectively.

Sensitivity coefficient for Δ T due to λ_{1}:

Sensitivity coefficient for Δ T due to λ_{2}:

Sensitivity coefficient for Δε due to λ_{1}:

Sensitivity coefficient for Δε due to λ_{2}:

Since the temperature is compensated, it is reasonable to assume there are non-correlated uncertainty components. The combined uncertainty is obtained from the uncertainties of the single components without taking into account possible covariances. The combined uncertainties for temperature and strain are the square root of

Thus the combined standard uncertainty for temperature difference as:

The combined standard uncertainty for strain difference as:

Effective degree of freedom for Δ T,

Effective degree of freedom for Δε,

Thus for coverage factor

Therefore, values of temperature and strain measurement uncertainty were determined to be 2.602 °C and 32.049 με, respectively. The estimation of expanded uncertainty provides at 95% confidence level with a coverage factor of

We present a simple and low-cost reference dual-wavelength grating sensor system that could offer the potential of simultaneous measurement of strain and temperature for infrastructures. Experimental results show that measurement errors of 6 με and 0.13 °C for strain and temperature could be achieved, respectively. We have performed and characterized the error analysis and measurement uncertainty for this strain-temperature sensing system. The maximum errors for temperature T and strain ε were calculated as 0.00155 T + 2.90 × 10^{−6} ε and 3.59 × 10^{−5} ε+ 0.01887 T, respectively. Based on the analysis of estimation of expanded uncertainty at 95% confidence level with a coverage factor of

The authors thank Yi-Hsien Wang for his laboratory assistance. The partial support of the National Science Council (NSC) of Taiwan under Contract Nos. NSC 95-2122-M-194-013, NSC 95-3114-P-194-001-MY3, NSC 96-2112-M-194-004-MY3, NSC 97-2111-E-224-050, and NSC 98-2111-E-224-060 is gratefully acknowledged.

A reference dual wavelength grating system.

Strain performance of grating sensor. Inset: output spectrum for the fiber Bragg grating sensors with an applied strain at 0 με (black color) and 900 με (magenta color), respectively.

Temperature performance of grating sensor. Inset: output spectrum for the fiber Bragg grating sensors with an applied temperature at 35 °C (black color) and 85 °C (magenta color), respectively.

3D Scatter plot of strain errors with applied temperature and applied strain.

3D scatter plot of temperature errors with applied temperature and applied strain.

Measured and calculated relative strain error as a function of strain. In error analysis, the measurement errors in wavelength were neglected.

Measured and calculated relative strain error as a function of strain. In error analysis, all measurement errors are taken into account.

Experimental and theoretical errors of individual strain and temperature measurement.

_{1}^{1} |
_{2}^{1} |
_{3} (1,551 nm) | |
---|---|---|---|

Strain coefficient (pm/με) | 0.914 ± 0.003 | 0.918 ± 0.003 | N/A^{2} |

Temperature coefficient (pm/°C) | 10.4 ± 0.10 | 12.1 ± 0.10 | 12.1 ± 0.08 |

Theoretical strain error (με) | 5.36 | 4.93 | N/A |

Experimental strain error (με) | 7.86 | 12.35 | N/A |

Theoretical temperature error (°C) | 0.17 | 0.17 | 0.17 |

Experimental temperature error (°C) | 0.65 | 0.44 | 0.48 |

Note:

Strain and temperature coefficients for calculation of measurement uncertainty were as follows: κ_{1ε} = 0.914 pm/με; κ_{1ε} = 0.918 pm/με; κ_{1T} = 10.4 pm/°C; κ_{2T} = 12.1 pm/°C

Not applicable