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The Comparation between Medium Capacitance and Medium Polarization Theory on Dielectric Constant
DOI: 10.12677/CMP.2021.103010, PDF, HTML, XML, 下载: 591  浏览: 917

Abstract: The phenomenon of the capacitance increasing by the medium has been explained by dielectric polarization up to now, or polar origin. The medium capacitance theory has been established after rock capacitance had been found from the phenomenon of the first and second voltage in induced polarization in geophysical exploration. The article compared the common ground and the difference between the two theories, which is very important to the theory of medium electrical property. The complex capacitance formula of medium capacitance theory, the Debye and Schweidler formula, has no difference mathematically. The charging and discharging formula of the capacitance origin is the same as Schweidler relaxation formula mathematically. There is no assumption in the conduction of the capacitance origin formula, but there is an assumption in the conduction of the polar theory formula. The assumption of formula conduction in the polar origin is actually the well-known fact of the voltage changing during a capacitor charging and discharging, which is adopted by capacitance origin formula conduction. The direction of charging discharging current in capacitance origin fits the fact, but not the polar origin. During the testing and calculating medium capacitance procedure using bridge system, the polar origin has in fact assumed capacitor shunted with a resistor. According to the capacitance origin, there will be no changes of medium capacitance with frequency changing in the modified bridge circuit by capacitance origin. The capacitance origin has explained the first and second voltage in the medium with no electric plates, but not the polar origin.

1. 引言

2. 介质电容理论

$Y=j\omega \frac{{C}_{1}}{1+j{R}_{1}{C}_{1}\omega }+j\omega \frac{{C}_{2}}{1+j{R}_{2}{C}_{2}\omega }+\cdots +j\omega \frac{{C}_{n}}{1+j{R}_{n}{C}_{n}\omega }+\frac{1}{{R}_{0}}$ (1)

$Y=j\omega {C}_{P}+j\omega \frac{{C}_{1}}{1+j{R}_{1}{C}_{1}\omega }+j\omega \frac{{C}_{2}}{1+j{R}_{2}{C}_{2}\omega }+\cdots +j\omega \frac{{C}_{n}}{1+j{R}_{n}{C}_{n}\omega }+\frac{1}{{R}_{0}}$ (2)

Cp为平行板电容为介质影响电场后的电容，其值为含介质电容频率趋于无穷的电容值。这可以理解为频率趋于无穷时，介质含内电阻电容的等效电容趋于0，电路的电容就是平行板电容Cp。式2可以写为

$Y=j\omega \left({C}_{p}+{\sum }_{i=1}^{n}\frac{{C}_{i}}{1+j{R}_{i}{C}_{i}\omega }+\frac{1}{j\omega {R}_{0}}\right)$ (3)

${C}_{i}={C}_{p}+{\sum }_{i=1}^{n}\frac{{C}_{i}}{1+j{R}_{i}{C}_{i}\omega }+\frac{1}{j\omega {R}_{0}}$ (4)

$I=±U\left(\frac{1}{{R}_{1}}\left({\text{e}}^{\frac{-t}{{R}_{1}{C}_{1}}}\right)+\frac{1}{{R}_{2}}\left({\text{e}}^{\frac{-t}{{R}_{2}{C}_{2}}}\right)+\cdots +\frac{1}{{R}_{n}}\left({\text{e}}^{\frac{-t}{{R}_{n}{C}_{n}}}\right)\right)$ (5)

3. Debye介质介电常数理论

Debye给出了介电常数和弛豫时间及频率关系的公式 [4]，根据该公式可以得到介电常数和损耗(电耗)的关系为一个半圆。Cole-Cole在该公式的基础上，给出一个圆弧的经验公式 [5] [6]，Davidson和Havriliak又提出了斜歪的圆弧的经验公式 [7]。现代对非均匀体的研究表明，介电常数和损耗的关系可以为多个圆弧。上述关系都有相应实验数据的支持 [8] [9]。Schweidler认为Debye公式偏离实验结果是因为介质极化并非是单一的，给出了复介电常数ε公式(Ai为权重因子)和弛豫函数 $f\left(t\right)$ 经验公式公式 [10]。

$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\left({\epsilon }_{0}-{\epsilon }_{\infty }\right){\sum }_{i}\frac{{A}_{i}}{1+j\omega {\tau }_{i}}$ (6)

$f\left(t\right)={\sum }_{i=1}^{n}\frac{{A}_{i}}{{\tau }_{i}}{\text{e}}^{\frac{-t}{{\tau }_{i}}}$ (7)

4. 介质电容理论与Debye介电常数理论对比

4.1. 公式对比

Debye的介电常数半圆公式是建立在假设的基础上的 [10]。即假设在t时刻介质极化强度Pr的增长速度dPr/dt正比于其终值 ${ϵ}_{0}{×}_{0}{E}_{0}$ 与该时刻Pr之差(1/τ为比例常数)。即

$\frac{\text{d}{P}_{r}}{\text{d}t}=\frac{1}{\tau }\left({ϵ}_{0}{×}_{r0}{E}_{0}-{P}_{r}\right)$ (8)

${C}_{i}={C}_{p}+{\sum }_{i=1}^{n}\frac{{C}_{i}}{1+j{R}_{i}{C}_{i}\omega }$ (9)

$\frac{\text{d}{V}_{c}}{\text{d}t}=\left({R}_{0}I-{V}_{c}\right)\left({R}_{0}+{R}_{1}\right)C$ (10)

Table 1. Given value of capacitor and resistor of dielectrics

Figure 1. Semicircle curve of equivalent Ce and periodic electrical loss Cϵ

Figure 2. Arc curve of equivalent Ce and periodic electrical loss Cϵ

Figure 3. Slope curve of equivalent Ce and periodic electrical loss Cϵ

Figure 4. Multimodal curve of equivalent Ce and periodic electrical loss Cϵ

$f\left(t\right)={\text{e}}^{-{\left(\frac{t}{\tau }\right)}^{\beta }}$ (11)

4.2. 放电电流方向

4.3. 介质电容测定

${ϵ}^{\prime }=\frac{C}{{C}_{0}}\frac{1}{1+{\omega }^{2}{R}^{2}{C}^{2}}$ (12)

${ϵ}^{″}=\frac{C}{{C}_{0}}\frac{\omega RC}{1+{\omega }^{2}{R}^{2}{C}^{2}}$ (13)

${C}_{i}=\frac{C}{1+jRC\omega }$ (14)

${C}_{i}=\frac{C}{1+{\omega }^{2}{R}^{2}{C}^{2}}$ (15)

${D}_{i}=\frac{{\omega }^{2}R{C}^{2}}{1+{\omega }^{2}{R}^{2}{C}^{2}}$ (16)

${C}_{p}={C}_{0}{\epsilon }_{\infty }$ (17)

${C}_{i}={C}_{0}\left({\epsilon }_{0}-{\epsilon }_{\infty }\right){A}_{i}$ (18)

${R}_{i}=\frac{{\tau }_{i}}{{C}_{i}}$ (19)

4.4. 无极板条件

4.5. 讨论

5. 结论

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