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压电陶瓷的内部极化状态随施加电压的变化而变化。当极化状态的变化与电压的变化不一致时,会导致电压上升和下降过程的曲线不一致,并存在反冲现象,称为磁滞现象。磁滞与材料、压电元件结构、电压变化和负载有关。除磁滞外,蠕变是另一个影响压电陶瓷定位精度的特性。由于极化后内部分子的摩擦力,变形无法及时完成,它需要一个滞后时间。当施加的电压保持不变时,位移会随时间缓慢变化,并在一段时间后可以达到稳定。当施加的驱动电压不同时,达到稳定性的时间也不同。由于压电陶瓷固有的磁滞和蠕变特性,可调F-P滤波器的膨胀和电压之间存在非线性关系,即透射峰的位置和F-P滤波器的电压是非线性的。为了解决FFP-TF的非线性问题,提出了以下FBG波长解调和波长优化算法。实际上,非线性激光波长调谐将导致测量误差和分辨率降低。
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FFPE的方案似乎相对简单,但是解调需要高精度和稳定性。理想的波长梳是由间隔隔开的一组离散的等距波长分量。但是,由于TSL的非线性,由采集卡收集的梳状滤波器峰值是不均匀的。对于光纤传感器的绝对测量,必须固定每个元件的位置,这可以通过先前的校准来完成。选择了日本横河电机的AQ6151光波长仪来校准FFPE的每个梳状滤波器峰值,波长计的精度高达±0.3 pm。通过AQ6151在15 min内测量44个梳状滤波器峰值的每个波长,并将相应的平均值设置为校准的NIR波长,该波长可用作测量的真实值,从而减少FFP-TF非线性的影响。因此,为了提高测量精度,使用线性插值法对FFPE的每个峰进行了连续校准,以校准每个采样点的波长。
一旦得出了上升扫描过程中FBG的时间,就可以直接确定每个FBG信号两侧的两个最近的FFPE梳。如图4所示,两个最接近的梳齿的采样时间分别为
${t_n}$ 和${t_{n + 1}}$ 。${\lambda _n}$ 和${\lambda _{n + 1}}$ 是对应于${t_n}$ 和${t_{n + 1}}$ 的参考通道的已知波长。在梳状滤波器峰值校准之后,将FFPE的光谱分为43个部分,分离出的光谱区域可用于线性查询。线性拟合可以提高解调的准确性,而FBG中心波长${\lambda _{{\rm{FBG}}}}$ 可以描述为:$${\lambda _{{\rm{FBG}}}} = {\lambda _n} + \frac{{{t_{{{\rm{FBG}}}}} - {t_n}}}{{{t_{n + 1}} - {t_n}}}({\lambda _{n + 1}} - {\lambda _n})$$ (1) -
以上方法仅削弱了FFP-TF非线性在解调系统中的影响。下面的算法可用于进一步优化解调数据。在综合监视系统中,可以将解调系统的信号假定为相对于采样点或时间((
${T_i}$ ,${\lambda _i}$ ),$i = 1,2\cdots , n$ )的记录波长数据的序列。将数据采样的步长设置为$\Delta T$ 时,将在瞬间生成第i个数据点:$${T_i} = \Delta T \cdot \left( {i - 1} \right)$$ (2) 第m个最小二乘拟合多项式为:
$$P({\lambda _i}) = \sum\nolimits_{k = 0}^{m - 1} {{c_k}T} _i^k$$ (3) 每个数据点的偏差为:
$$\left. {\left| {{\varepsilon _i}} \right.} \right| = \left. {\left| {{\lambda _i} - P({\lambda _i})} \right.} \right|$$ (4) 为了最小化偏差平方的总和,使用多项式最小二乘曲线拟合数据:
$$ S={\sum\limits_{i=1}^{n}({\lambda }_{i}}{{-P}({\lambda }_{\rm{i}}))}^{2}=\sum\limits_{i=1}^{n}\left({\lambda }_{i}-{{\sum }_{k=0}^{m-1}{c}_{k}}{T}_{i}^{k}\right)^{2}$$ (5) 当公式(5)结果达到最小值时,多个函数值条件指示:
$$\frac{{\partial S}}{{\partial {C_k}}} = 2\sum\limits_{i = 1}^n {\left[ {{\lambda _i} - \sum\nolimits_{k = 1}^n {{C_k}T_i^k} } \right]} \cdot T_i^k$$ (6) 在公式(6)中引入内积简化后,可以得到多项式(7):
$$A = \left[ {\begin{array}{*{20}{c}} {{C_1}} \\ \ldots \\ {{C_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} n\;{\displaystyle\sum\limits_{i = 1}^n {{T_i}} }\; \ldots \;{\displaystyle\sum\limits_{i = 1}^n {T_i^m} } \\ {\displaystyle\sum\limits_{i = 1}^n {{T_i}} }\;{\displaystyle\sum\limits_{i = 1}^n {T_i^2} }\; \ldots \;{\displaystyle\sum\limits_{i = 1}^n {T_i^{m + 1}} } \\ {\displaystyle\sum\limits_{i = 1}^n {T_i^m} }\;{\displaystyle\sum\limits_{i = 1}^n {T_i^{m + 1}} }\; \ldots \;{\displaystyle\sum\limits_{i = 1}^n {T_i^{2m}} } \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\displaystyle\sum\limits_{i = 1}^n {{\lambda _i}} } \\ {\displaystyle\sum\limits_{i = 1}^n {{T_i}{\lambda _i}} } \\ {\displaystyle\sum\limits_{i = 1}^n {T_i^m{\lambda _i}} } \end{array}} \right]$$ (7) 然后获得数据点
$\left( {{T_i},P({\lambda _i})} \right)\left( {i = 1,2 \cdots n} \right)$ 的值。均方根为${\varepsilon _x} = \sqrt {\dfrac{{\displaystyle\sum\limits_{i = 1}^n {{{\left( {{\lambda _i} - P({\lambda _i})} \right)}^2}} }}{{n - 1}}} $ 。根据拉吉达准则,将总错误阈值设置为$3{\varepsilon _x}$ ,相应的置信度为97%。$$\left. {\left| {{\lambda _i} - P({\lambda _i})} \right.} \right| > 3{\varepsilon _x}$$ (8) 满足公式(8)的数据是总误差,需要消除然后替换。当需要替换单个点时,
${\lambda _i}$ 是一个严重错误,替换值为:$${\lambda _i} = \frac{{{\lambda _{i - 1}} + {\lambda _{i - 2}} + {\lambda _{i + 2}} + {\lambda _{i + 1}}}}{4}$$ (9) 当需要对附近的两个点进行插值时,都是严重误差,则替换值为:
$${\lambda _i} = \frac{{{\lambda _{i - 1}} + {\lambda _{i - 2}} + {\lambda _{i + 2}}}}{3}$$ (10) 粗差处理的流程图如图5所示。将参考部分中的初始值设置为整个粗略误差数据估计的第一值,然后估计下一点的状态值。比较估计值和实际值之间的差异,如果该差异较小,则估计该估计值是合理的。否则,该点被认为是严重错误,需要更换。当参考零件向后移动一个点时,其第一个值将被删除,并由参考零件中的相邻值代替。如此循环直到数据处理完成,然后继续下一步。在上一轮中替换的数据将确定该轮数据中是否存在新的总差额。如果不是,则计算完成。否则,循环执行上述操作。
High-precision FBG demodulation system using near-infrared wavelength scanning laser
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摘要: 光纤布拉格光栅(FBG)由于其轻巧、规模小、不受电磁干扰和复用能力的影响等优点,广泛用于监视结构健康、机械运行、航空航天和其他领域。引入可调谐扫描激光(TSL)来研制近红外(NIR)范围内的精确光纤布拉格光栅(FBG)波长解调系统,实现高速度、宽范围、高精度的解调。采用一种光纤法布里-珀罗标准具(FFPE)用作波长标记以提取波长在细分波长扫描范围内实现分段线性解调,解决可调扫描激光器带来的非线性问题。引入了另一种光纤法布里–珀罗标准具,实现解调的高精度校准。提出一种多项式最小二乘曲线拟合算法,进一步提高解调的准确性和稳定性,利用了波长范围为1525~1565 nm的近红外波长扫频激光器,得到了非常优异的结果,解调系统的精度为±0.5 pm,实现了高精度、简易化和小型化。
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关键词:
- FBG解调 /
- 光纤传感 /
- 近红外 /
- 可调谐扫描激光 /
- 光纤法布里-珀罗标准具
Abstract: Fiber Bragg grating (FBG) has been widely used in monitoring structural health, mechanical operation, aerospace field, and other physical parameters due to its advantages of being lightweight, tiny scale, immune to electromagnetic interference and multiplexing capability.Tunable scanning laser (TSL) was introduced to fabricate an accurate FBG wavelength demodulating system in the near infrared range (NIR) to achieve high-speed, wide-range, and high-precision demodulation. One fiber Fabry–Perot etalon (FFPE) was used as a wavelength marker to extract wavelength. The wavelength scanning range was subdivided to implement piecewise linear demodulation, which solves the nonlinear problem brought about by tunable scanning laser. Another fiber Fabry–Perot etalon was introduced to realize high-accuracy calibration of the demodulation. A polynomial least square curve fitting algorithm was proposed to further enhance demodulation accuracy and stability. This work has been utilizing a near-infrared wavelength scanning laser with a wavelength range of 1525-1565 nm. It showed a very promising result whereby the accuracy of the demodulating system exceeds ±0.5 pm, which achieves high accuracy, simplification and miniaturization. -
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