# Dielectric Characterization of Chinese Standard Concrete for Compressive Strength Evaluation
## 3 Experimental Results
### Dielectric Constant Modelling
#### 3.3.3 Jonscher Model  
The Jonscher model uses complex electric susceptibility (\(\chi_{c}\)) to describe the sum of various susceptibilities related to polarizations responding to an applied electric field. A real part of the relative effective permittivity at the very high frequency (\(\varepsilon_{\infty}\)), was used to describe the limiting low value of the dielectric constant, whereas the complex permittivity holds a format of \(\varepsilon_{r}\)\({}^{\prime}-j\varepsilon_{r}\)\({}^{\prime\prime}\), as given by [40; 41]:  
\[\varepsilon_{r}^{\prime}(f)=\chi_{r}\left(\frac{f}{f_{r}}\right)^{n-1}+ \varepsilon_{\infty} \tag{7}\]  
\[\varepsilon_{r}^{{}^{\prime\prime}}(f)=\chi_{r}\left(\frac{f}{f_{r}}\right)^{ n-1}\cot\left(\frac{n\pi}{2}\right) \tag{8}\]  
where  
* \(n\) is an empirical parameter in the range of 0 to 1 for modelling the dielectric loss;
* \(\chi_{r}\) is the real part of \(\chi_{c}\);
* \(f_{r}\) is an arbitrarily chosen reference frequency. In this study, we picked \(f_{r}=0.1\) GHz. The three dielectric constant regression models all involve three fitting parameters, respectively, as expressed in Equations (4)-(8). Although the exponential model only gives a real-part description, its frequency response is found to be very similar to that from the Debye model. Figure 6 shows curve fitting of the three models, in both the linear- and log-frequency scale. The use of the log-scale gives a clear insight into the regressions at low frequencies, whereas the linear scale plots provide the fit conditions at high frequencies, viz., 2 to 6 GHz. While dielectric permittivity models were only available in the literature up to 1 or 2 GHz, e.g., [29; 41], the dielectric dispersion of concrete at higher frequencies is therefore the focus in this study. This is owing to the current popularity of wireless communication signals, such as the 4G cellular phone signals and the indoor WiFi signals, whose penetration abilities through concrete walls very much depend on the dielectric properties. From Figure 6b, it is observed that, in general, the Jonscher model provides the best fit to the measured data in all cases, when compared to the other models, whereas the Debye model identifies the same trends of decay as the exponential model. By comparing expressions (4) and (5), one can express the relation of the time constant (\(b\)) in terms of the relaxtion time (\(\tau\)), as shown below. Figure 6: Dielectric characterization of GB concrete via curve fitting to measured data using the first-order exponential model, Debye model, and Jonscher model, expressed in (**a**) linear scale; (**b**) log-frequency scale. With \(C\) equals \(\varepsilon_{\infty}\) (see Tables 4 and 5):  
\[Ke^{-fb}=\frac{\Delta\varepsilon}{1+\left(2\pi f\tau\right)^{2}}\, \tag{9}\]  
\[b\cong\frac{2}{f}ln(2\pi f\tau)-\frac{1}{f}ln\left(\frac{\Delta\varepsilon}{K} \right)\ for\ (2\pi f\tau)^{2}\gg 1 \tag{10}\]  
An interesting finding has been observed, where the limiting values of the dielectric constant increase in order with the GB concrete types. The mathematical relations of the dielectric constant, compressive strength, and initial water contents of GB concrete, will be further examined and discussed in the next section. All regression coefficients (with 95% confidence bounds) and goodness-of-fits of the three models are summarized in Tables 4-6, respectively. The goodness-of-fit was obtained by using the curve-fitting tools in MATLAB, which are represented by the _sum of squares due to error_ (_SSE_) and the _R-square_ (\(R^{2}\)) [42]. These two indications are interrelated, and are sufficient to validate the regression model by using any values between 0 and 1. When \(R^{2}=1\) and \(SSE=0\), the proposed regression model is indicated as a perfect fit for the observation data. Based on the values of the goodness-of-fit listed in Tables 4-6, the authors can justifiably state that the Jonscher model outperforms the other models for the whole frequency range of 0.1 to 6 GHz. This corroborates the previous finding reported in [40; 41].