## 1 Introduction
### Complex Dielectric Permittivity
\begin{split}&\mathrm{Im}\Big{[}\hat{\varepsilon}^{(\omega;2 \omega;3\omega)}\Big{]}=\varepsilon_{\infty}\Bigg{(}\frac{\Gamma_{2}^{( \omega)}}{\left(1-\Gamma_{1}^{(\omega)}\right)^{2}+\left(\Gamma_{2}^{(\omega)} \right)^{2}}+\mathrm{Y}\mathrm{E}_{0}^{2}\times\\ &\times\Big{(}\chi_{1}^{(\omega;2\omega;3\omega)}\sin(2\omega \mathrm{t})-\chi_{2}^{(\omega;2\omega;3\omega)}\cos(2\omega\mathrm{t})\Big{)} \Big{)}.\end{split}  \tag{108b}\]  
The current density vector is calculated by taking into account the nonlinearity of the CDP at the set of frequencies \(\Omega=\{\omega;2\omega;3\omega;\ldots;2\lambda\omega;(2+1)\omega;\ldots\}\):  
\[\overset{\rightarrow}{j}^{(\Omega)}(\mathrm{t})=\sigma\overset{\rightarrow}{ \mathrm{E}}_{\mathrm{pol}}(\mathrm{t})+\frac{\partial\overset{\rightarrow}{D} ^{(\Omega)}(\mathrm{t})}{\partial t}. \tag{109}\]  
By incorporating into (109)  
\[\overset{\rightarrow}{D}^{(\Omega)}(\mathrm{t})=\varepsilon_{0}\hat{ \varepsilon}^{(\Omega)}\overset{\rightarrow}{\mathrm{E}}_{\mathrm{pol}}( \mathrm{t}), \tag{110}\]  
in the case of \(\overset{\rightarrow}{\mathrm{E}}_{\mathrm{pol}}(\mathrm{t})=\overset{ \rightarrow}{\mathrm{E}}_{0}\mathrm{exp}(\mathrm{i}\mathrm{t})\), taking into account (97), we obtain  
\[\overset{\rightarrow}{j}^{(\Omega)}(\mathrm{t})=\sigma\overset{\rightarrow}{ \mathrm{E}}_{\mathrm{pol}}(\mathrm{t})+\mathrm{i}\omega\varepsilon_{0} \varepsilon_{\infty}\overset{\rightarrow}{\mathrm{E}}_{\mathrm{pol}}(\mathrm{t })+\mathrm{i}\omega\overset{\rightarrow}{\sum}_{\lambda=0}^{\infty}(2\, \lambda+1)\varepsilon_{0}\delta_{2\lambda}^{(\Omega_{2\lambda+1})}\overset{ \rightarrow}{\mathrm{E}}_{\mathrm{pol}}^{2\lambda+1}(\mathrm{t}). \tag{111}\]  
Conduction current \(\overset{\rightarrow}{j}_{\mathrm{cond}}(\mathrm{t})=\sigma\overset{ \rightarrow}{\mathrm{E}}_{\mathrm{pol}}(\mathrm{t})\) due to _the through_ movement of charge carriers (ions; in the HBC, protons), according to the experimental results, significantly affects the current density \(\overset{\rightarrow}{j}^{(\Omega)}(t)\) only in high-temperature regions (350-450 K) [1,3,4], with _the nonlinear_ properties of the _relaxation_ motion of protons reflected in the third term of the Formula (111). The function \(\overset{\rightarrow}{j}_{\text{cond}}(t)\) is calculated in a linear approximation over the field \(\overset{\rightarrow}{\text{E}}_{\text{pol}}(t)\). From (110), we have  
\[ \begin{split}\text{Re}\left[\overset{\rightarrow}{j}^{(\Omega)}(t )\right]=\sigma\overset{\rightarrow}{\text{E}}_{0}&\text{cos}( \omega t)-\omega\varepsilon_{0}\varepsilon_{\omega}\overset{\rightarrow}{ \text{E}}_{0}&\text{sin}(\omega t)+\omega\overset{\infty}{\sum }_{\lambda=0}^{\infty}(2\lambda+1)\varepsilon_{0}\overset{\rightarrow}{\text {E}}_{0}&\left\{\text{Im}\left[\hat{\varepsilon}_{2\lambda}^{( \Omega_{2\lambda+1})}\right]\right.\times\\ &\times\text{cos}((2\lambda+1)\omega t)-\text{Re}\left[\hat{ \varepsilon}_{2\lambda}^{(\Omega_{2\lambda+1})}\right]\times\text{sin}((2 \lambda+1)\omega t)\right\}&,\\ \text{Im}\left[\overset{\rightarrow}{j}^{(\Omega)}(t)\right]= \sigma\overset{\rightarrow}{\text{E}}_{0}&\text{sin}(\omega t )+\omega\varepsilon_{0}\varepsilon_{\omega}\overset{\rightarrow}{\text{E}}_{0}& \text{cos}(\omega t)+\omega\overset{\infty}{\sum}_{\lambda=0}^{\infty}(2 \lambda+1)\varepsilon_{0}\overset{\rightarrow}{\text{E}}_{0}& \left\{\text{Re}\left[\hat{\varepsilon}_{2\lambda}^{(\Omega_{2\lambda+1})} \right]\right.\times\\ &\times\text{cos}((2\lambda+1)\omega t)-\text{Im}\left[\hat{ \varepsilon}_{2\lambda}^{(\Omega_{2\lambda+1})}\right]\times\text{sin}((2 \lambda+1)\omega t)\right\}&.\end{split}  \tag{111b}\]  
After introducing the complex function  
\[\overset{\rightarrow}{\text{E}}_{\text{pol}}(t)=\sum_{\lambda^{\prime}=0}^{ \infty}\overset{\rightarrow}{\text{E}}_{0}^{\rightarrow 2\lambda^{\prime}+1} \exp\Bigl{(}\text{i}\left(2\lambda^{\prime}+1\right)\omega t\Bigr{)} \tag{112}\]  
using (111a) and (111b), we construct the following expressions:  
\[