## 2 Materials and Methods  
### Dielectric Constant Extraction Method  
The dielectric constants of materials can be obtained by employing the scattering parameters and four conversion techniques: Nicholson-Ross-Weir (NRW), NIST iterative, new non-iterative, and short circuit line [16]. Among these methods, the NRW method is popularly used to derive the dielectric constant of finite thickness material by employing two or all scattering parameters depending on the measurement technique and some mathematical manipulation [17,18]. A challenge encountered when characterizing the electromagnetic properties is the error resulting from the numerical singularity that exists when the sample thickness is equal to an integer multiple of half the wavelength. In our study, we utilize only S21 as the initial data for extracting the relative permittivity and  
Figure 1: Block diagram of (**a**) the conventional method and (**b**) the proposed method.  
loss tangent of low-loss non-magnetic materials as input features of the training model. Therefore, the NIST iterative method seems to be applicable compared to the others since it largely limits the mentioned error of the NRW method and it was proven to be the most conformable method for this type of material [19]. The reflection coefficient, \(\Gamma\), can be obtained from  
\[\Gamma=\frac{\frac{\gamma_{0}}{\mu_{0}}-\frac{\gamma}{H}}{\frac{\gamma_{0}}{\mu_ {0}}+\frac{\gamma}{H}} \tag{1}\]  
where the \(\gamma_{0}\) is the propagation constant in air, which can be defined as:  
\[\gamma_{0}=j\sqrt{\left(\frac{\omega}{c}\right)^{2}-\left(\frac{2\pi}{\lambda _{c}}\right)^{2}} \tag{2}\]  
and \(\gamma\) is the propagation constant in MUT defined as:  
\[\gamma=j\sqrt{\frac{\omega^{2}\mu_{r}\epsilon_{r}}{c^{2}}-\left(\frac{2\pi}{ \lambda_{c}}\right)^{2}}=j\sqrt{\epsilon_{r}\epsilon_{0}\mu_{0}\omega^{2}- \left(\frac{2\pi}{\lambda_{c}}\right)^{2}} \tag{3}\]  
where:  
* \(\mu=\mu_{r}\mu_{0}\) and \(\epsilon=\epsilon_{r}\epsilon_{0}\) with \(\mu_{r}\) and \(\epsilon_{r}\) are the relative permeability and permittivity, respectively, and \(\mu_{0}\) and \(\epsilon_{0}\) are the vacuum permeability and permittivity, respectively;
* \(c=1/\sqrt{\mu_{0}\epsilon_{0}}\) is the velocity of light;
* \(\omega=2\pi f\) is the angular frequency, with \(f\) as the frequency of the signal;
* \(\lambda_{c}\) is the cutoff wavelength.  
Since the MUTs are non-magnetic materials, \(\mu_{r}=1\). Then, the reflection coefficient in (1) can be rewritten as:  
\[\Gamma=\frac{\gamma_{0}-\gamma}{\gamma_{0}+\gamma} \tag{4}\]  
The transmission coefficient is:  
\[T=e^{-\gamma d}=e^{(-jd\sqrt{\varepsilon_{r}\epsilon_{0}\mu_{0}\omega^{2}- \left(\frac{2\pi}{\lambda_{c}}\right)^{2}})} \tag{5}\]  
with \(d\) as the MUT thickness.  
By solving Equations (4) and (5), the relative permittivity can be determined as:  
\[F(\epsilon_{r})=\frac{S_{21}+S_{12}}{2(1-T^{2}\Gamma^{2})}-T\Big{(}1-\Gamma^{ 2}\Big{)}e^{-j\gamma_{0}\left(L_{air}-d\right)} \tag{6}\]  
The relative complex dielectric permittivity can then be computed from the scattering parameters by solving Equation (6) using the Newton-Raphson iterative approach [19; 20]. The algorithm is considered converged when \(F(\epsilon_{r})=0\).