## 2 Materials and Methods
### EMI Shielding Model  
The longitudinal and transverse conductivity estimated from the equivalent circuit models explained above is used to predict the EMI SE of hybrid BP/CFRP laminates. The EMI shielding in materials is a consequence of three fundamental phenomena: reflection, absorption, and internal reflections. In the case of composites, the absorption-based shielding effect exceeds 10 dB; hence, the effect of shielding through multiple reflections is safely ignored. The EMI SE of the samples was determined using an ANSYS high-frequency structure simulator (HFSS), which uses a full-wave electromagnetic simulation technique to calculate the scattering parameters (S-parameters). The software employs the finite element boundary integral method (FEBIM) to solve [PERSON]'s equations that govern electromagnetic fields within each mesh element and account for interaction with exterior regions. Although the constituent properties employed for the EMI simulations are obtained from the micro-scale model explained in Section 2.1, the electromagnetic simulations themselves are conducted at the lamina scale and do not address the effects of multiple scattering at the constituent scale. For this reason, experimental testing using an appropriate setup and vector network analyzer is required to validate the ANSYS simulations.  
In the modeling of each lamina, the material properties, including conductivity, permittivity, and permeability, are provided as inputs. The electrical conductivity for a lamina oriented at an arbitrary angle \(\theta\) is represented in tensor form, as depicted in Equation (22) [24]. \(\sigma_{L}\) and \(\sigma_{T}\) denote the longitudinal and transverse conductivities. Due to the transversely isotropic nature of the lamina, the through-ply conductivity is assumed to be the same as the transverse conductivity. A semi-infinite representative shield element was simulated, and the scattering parameters (S-parameters) S\({}_{11}\) and S\({}_{21}\) were obtained for each sample under study, which allow for calculating the amplitude of the reflected and transmitted wave. A pictorial representation of a 10 mm \(\times\) 10 mm rectangular waveguide used for the analysis is shown in Figure 10. To prevent boundary reflections/radiations and ensure precise results, periodic boundary conditions were applied along the walls, and Floquet ports [24] were employed at both the Inlet and Exit ports. The fundamental transverse electric mode of propagation TE\({}_{10}\) was considered for the analysis due to the simplicity of the geometry and nonmagnetic nature of the specimen. This approach allowed for a comprehensive analysis of the electromagnetic response of the representative shield element under investigation.  
\[\sigma_{\theta}=\begin{bmatrix}\sigma_{L}cos^{2}(\theta)+\sigma_{T}sin^{2}( \theta)&\frac{\sigma_{L}-\sigma_{T}}{2}sin(2\theta)&0\\ \frac{\sigma_{L}-\sigma_{T}}{2}sin(2\theta)&\sigma_{L}sin^{2}(\theta)+\sigma_ {T}cos^{2}(\theta)&0\\ 0&0&\sigma_{T}\end{bmatrix} \tag{22}\]  
Using S-parameters obtained from the simulations, the shielding effectiveness due to reflection \(SE_{R}\) and absorption \(SE_{A}\) were calculated using the following relations:  
\[SE_{R}=10 log_{10}\left(\frac{1}{1-\left|\xi_{11}^{2}\right|}\right)\,\left[ \text{dB}\right] \tag{23}\]\[SE_{A}=10 log_{10}\left(\frac{1-\left|\xi_{11}^{2}\right|}{\left|\xi_{21}^{2} \right|}\right)\ [\text{dB}] \tag{24}\]  
Finally, the total shielding effectiveness \(SE\) can be expressed as follows:  
\[SE=SE_{A}+SE_{R} \tag{25}\]