非均匀非对称情况下平面电容传感器的数学模型及应用研究

Mathematical Model and its Application of the Planar Capacitance Sensor under Non-uniform and Non-symmetrical Conditions

针对利用平面电容传感器测量木材含水率这一工程问题,本文给出一个能体现电介质介电常数分布情况与平面电容传感器电容量之间关系的数学模型,这是在非均匀非对称条件下建立的,它是带有连接条件和边界积分条件的三维Laplace方程.同时,本文讨论了该模型解的唯一性,并且以均匀对称情况为例,运用有限差分法对算例进行计算,分析结果并验证了该模型的合理性.由于木材等介质的含水率直接影响其介电常数,所以通过电容传感器测量电容值,可以得到木材等介电常数的分布情况,进而得到木材含水率的分布情况.该模型可应用到上述问题等更深层的实际问题中.

In this paper, we propose a new mathematical model for the engineering problem of measuring the timber moisture content by a planar capacitance sensor. It could reflect the relation between the distribution of timber＇s dielectric permittivity and the capacitance of a planar capacitance sensor, and was established under non-uniform and non-symmetrical condi- tions. The model is a three-dimensional Laplace equation with joint conditions and boundary integral connections. At the same time, we discussed the uniqueness of solution to this equa- tion. Then an example was promoted in symmetrical conditions and calculated by the finite difference method. We analyzed the results and verified the rationality of the model. We found that the moisture content of timber directly influence the dielectric constant distribution of timber. Therefore, as long as we get the distribution of the dielectric constant, by measuring the capacitance value of a capacitance sensor, we could get the moisture content. The proposed model can be applied to other practical problems.