## Methodology
### The Finite Element Method  
In this work, the FEM was used to study the MA properties of C-Fe\({}_{3}\)O\({}_{4}\)/PVDF in a rectangular waveguide. In setting up the geometry, the WR90 and WR62 rectangular waveguides were modeled in 3D with schematic shown in Figure 2. The complex permeability and permittivity value from measurement with VNA were used in simulating the MA properties. All boundaries of the waveguide were simulated as a perfect electric conductor except two boundaries marked as port 1 and port 2 for wave excitation. The geometry has 3 domains, the first and third domain was modelled as free space while the second domain contains the material. Then a mesh consisting of tetrahedral elements was generated. The complete mesh consists 10,574 domain elements, 2256 boundary elements, and 264 edge elements. The EM propagating mode used in this study was the TE\({}_{10}\) mode. The fundamental TE\({}_{10}\) is the mode with the lowest cut-off frequency that can propagate through a rectangular waveguide. Then, the scattering parameters were evaluated at the position S1 and S2, respectively and used in calculating the MA properties. In this model, the radio frequency (RF) module's port boundary condition was used for the wave excitation. With this boundary condition, the magnitude of the reflection/transmission and reflection loss can be estimated [16]. The wave equation of the propagating electric field \"E\" based on the Maxwell equation is given below. \[\
abla\times\mu_{\mathrm{r}}^{-1}(\
abla\times\mathrm{E})-k_{0}^{2}\bigg{(} \varepsilon_{\mathrm{r}}-\frac{\mathrm{j}\sigma}{\omega\varepsilon_{0}}\bigg{)} \mathrm{E} \tag{2}\]  
where \(\varepsilon_{\mathrm{r}}\), \(\mu_{\mathrm{r}}\), \(\sigma\), \(\omega(2\pi\mathrm{f})\), \(\mathrm{f}\), \(\varepsilon_{\mathrm{o}}\), \(\mathrm{k}\) are the relative permittivity, relative permeability, conductivity, frequency, permittivity of free space and wave vector, respectively. The E-field within the waveguide was discretized by means of tetrahedron elements [16]. Therefore, within each element, the EM field can be interpolated from the individual node value with the first order polynomial as follows:  
\[\rho^{\mathrm{e}}(\mathrm{x},\mathrm{y},\mathrm{z})=\mathrm{a}^{\mathrm{e}}+ \mathrm{b}^{\mathrm{e}}\mathrm{x}+\mathrm{c}^{\mathrm{e}}\mathrm{y}+\mathrm{d} ^{\mathrm{e}}\mathrm{z} \tag{3}\]  
The E-Field in the rectangular waveguide using FEM is  
\[\mathrm{E}^{\mathrm{e}}=\sum\
olimits_{\mathrm{i}=1}^{6}N_{\mathrm{i}}^{ \mathrm{e}}E_{\mathrm{i}}^{\mathrm{e}}(\mathrm{X}.\mathrm{Y}.\mathrm{Z}) \tag{4}\]  
where \(N_{\mathrm{i}}^{\mathrm{e}}\), \(\mathrm{i}=1\), \(2\), \(3\), \(\ldots\), \(6\) are the complex intensities of the E-Field correlated with six edges of the element, and \(E_{\mathrm{i}}^{\mathrm{e}}(\mathrm{x},\mathrm{y},\mathrm{z})\) is the vector basis function related to the \(\mathrm{i}\)-th edge of the element. Applying boundary conditions, the integration over the volume of one element is:  
\[ \begin{array}{c}\frac{1}{\mu_{\mathrm{r}}}\left(\sum_{\mathrm{i}=1}^{6}N_{ \mathrm{i}}^{\mathrm{e}}\iint\left(\
abla\times\overrightarrow{W_{\mathrm{ i}}}\cdot\
abla\times\overrightarrow{W_{\mathrm{i}}}-K_{0}^{2}\varepsilon_{ \mathrm{r}}\overrightarrow{W_{\mathrm{i}}}\cdot\overrightarrow{W_{\mathrm{ j}}}\right)\mathrm{d}\mathrm{v}2\left(\frac{\mathrm{i}\
u\mathrm{i}}{\mu_{\mathrm{r}}} \right)\cdot Y_{0}^{\mathrm{i}}\iint\overrightarrow{W_{\mathrm{j}}}\cdot \overrightarrow{e_{\mathrm{o}}}(\mathrm{x},\mathrm{y})\mathrm{d}\mathrm{s} \right.\\ \left.-\left(\frac{\mathrm{i}\
u\mathrm{i}}{\mu_{\mathrm{r}}}\right)\sum_{ \mathrm{i}=1}^{6}N_{\mathrm{i}}^{\mathrm{e}}\sum_{\mathrm{p}=0}^{\infty}Y_{ \mathrm{p}}^{\mathrm{f}}\left(\iint\overrightarrow{W_{\mathrm{j}}}\cdot \overrightarrow{e_{\mathrm{p}}}(\mathrm{x},\mathrm{y})\mathrm{d}\mathrm{s} \iint\overrightarrow{W_{\mathrm{i}}}\cdot\overrightarrow{e_{\mathrm{p}}}( \mathrm{x},\mathrm{y})\mathrm{d}\mathrm{s}\right)\right)\end{array}