Kramers-Kronig变换在介电响应分析中的数值计算方法、意义及应用

Kramers-Kronig关系是因果系统中的基本物理规律,在电介质物理领域,Kramers-Kronig关系描述的是极化率的实部与虚部之间的内在数学联系。实际测量获得的有限频段离散频率点上的介电响应复电容结果无法直接使用Kramers-Kronig关系的数学表达式进行计算,且由于无穷频率电容和可能的电导过程的存在,复电容的实部和虚部一定不满足Kramers-Kronig关系。提出基于普适介电响应理论的数据外延和插值方法,有效地扩大了Kramers-Kronig变换数值计算的积分范围,并对整个频域进行分段划分,在各个频段上分别实现Kramers-Kronig关系的解析计算或数值积分计算,实现了针对实际测量获得的有限频段离散频率点上的介电响应的Kramers-Kronig变换数值计算。无穷频率电容项和电导项通过Kramers-Kronig变换后的结果为0,因此,通过Kramers-Kronig变换及对比分析,可以实现对复电容中的极化信息、电导信息和无穷频率电容信息的解耦分析。在高温硫化硅橡胶的介电响应中,通过Kramers-Kronig变换数值计算和后续的对比分析,可以剔除其中的电导和无穷频率电容的影响,进而观察到低频段的低频弥散现象和高频段的实部与虚部相互平行的指数规律。

Numerical Computational Method,Application and Significance of the Kramers-Kronig Transform in the Analysis of Dielectric Response

The Kramers-Kronig relation is the basic physical law in the causal system.In the field of dielectric physics,the Kramers-Kronig relation describes the intrinsic mathematical connection between the real and imaginary components of the susceptibility.The dielectric complex capacitance obtained by actual measurement cannot be directly calculated using the original mathematical expression of the Kramers-Kronig relation,furthermore,the real and imaginary components of the complex capacitance do not obey the Kramers-Kronig relation due to the existence of infinite frequency capacitance and possible conductivity.The present work proposed a data extrapolation and interpolation method based on the universal dielectric relaxation law,which effectively expands the integral range of the Kramers-Kronig transform numerical calculation.Then,the whole frequency domain were divided into subdomains,in each subdomain,the Kramers-Kronig relation can be calculated analytically or numerically.The Kramers-Kronig transform numerical calculation for the dielectric response at the discrete frequency points within the finite frequency domain was realized by superimposing these results in all subdomains.The results of the Kramers-Kronig transform of the infinite frequency capacitance term and the conductivity term are zero,therefore,the decoupling analysis of the polarization information,the conductivity information and the infinite frequency capacitance information in the measured dielectric complex capacitance can be realized by this Kramers-Kronig transform numerical calculation and the relevant comparative analysis.In the dielectric response of high temperature vulcanized silicone rubber,through the numerical calculation of Kramers-Kronig transform and subsequent comparative analysis,after eliminating the influence of conductance and infinite frequency capacitance,the low frequency dispersion phenomenon in the low frequency range and the parallel power law in the high frequency range can be observed.