## 2 Materials and Methods
### Electrical Conductivity Characterization of CFRPs
#### 2.1.1 Longitudinal Conductivity  
For the longitudinal case, the analysis is based on the premise that in addition to their internal electrical resistance, the fibers also possess inductive effects, while the matrix acts as an insulating material. Figure 2 presents a graphical depiction of a unidirectional (UD) CFRP lamina, showcasing its microstructure and the associated equivalent circuit for the longitudinal orientation. As depicted, individual fibers are represented as a combination of a resistor (R) and an inductor (L) arranged in series. For the N parallel fibers, the overall circuit consists of N parallel resistor-inductor elements. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter (Unit)** & **Value** \\ \hline Fiber mean diameter (\(\mu\)m) & 5.24 \\ Fiber diameter standard deviation (\(\mu\)m) & 0.295 \\ Fiber volume fraction & 0.6 \\ CFRP lamina thickness (mm) & 0.25 \\ BP thickness (mm) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relevant composite sample parameters used in the study. Figure 2: Microstructure of a unidirectional CFRP lamina and its equivalent electrical circuit in the longitudinal direction consisting of the \(N\) fibers, each having their own resistance (\(R_{f}\)) and inductance (\(L_{f}\)). The resistance of each wire \(R_{f}\) can be expressed in terms of fiber resistivity \(\rho_{f}\), length \(len_{f}\), and radius \(r_{f}\) as follows:  
\[R_{f}=\frac{\rho_{f}len_{f}}{\pi r_{f}^{2}} \tag{1}\]  
The inductance of each fiber, denoted \(L_{f}\), is expressed as a sum of its self-inductance and the mutual inductance of the neighboring fibers. The self inductance represented as \(\left(L_{f}\right)_{S}\) can be expressed in terms of the fiber dimensions and vacuum permeability \(\mu_{o}\)[34]:  
\[\left(L_{f}\right)_{S}=\frac{\mu_{o}len_{f}}{2\pi}log\left(\frac{2 len_{f}}{r_ {f}}-1\right) \tag{2}\]  
For mutual inductance estimation, drawing upon the principles of [PERSON]'s law of electromagnetic induction and [PERSON]'s law when an alternating current flows through two parallel wires separated by a distance d, a counteracting current is induced. This happens because the changing current gives rise to a changing magnetic field through the surrounding wires. Consequently, a current is generated that resists this magnetic field change. This phenomenon within the CFRP microstructure is depicted in Figure 3a. The direction of the magnetic field can be determined using the right-hand rule, and its magnitude can be expressed as below:  
\[B=\frac{\mu_{o}I}{2\pi d} \tag{3}\]  
where \(I\) is the electric current traversing the fiber and \(d\) denotes the radial distance from the fiber to the point of interest. Figure 3b depicts magnetic field vectors within a small region of the py cross-section and illustrates the influence of surrounding fibers on the magnetic field intensity at any given fiber location. For illustration, the fibers are color-coded to correspond with their respective magnetic field vectors, with an 'X' symbol denoting the direction of current flow into the plane. It should be noted that the effective magnetic field at the arbitrary fiber experiences significant cancellation due to the influence of surrounding fibers. However, it remains nonzero owing to the nonuniform distribution of fibers within the microstructure. Finally, using the relation between magnetic flux \(\varphi\) and wire inductance \(L\), the final form of mutual inductance denoted by \(\left(L_{f}\right)_{M}\) can be expressed as follows:  
\[\left(L_{f}\right)_{M}=\frac{\varphi}{I}=\frac{B.\text{A}}{I}=\frac{2 Blen_{f} r_{f}}{I} \tag{4}\]  
Figure 3: (**a**) Magnetic fields due to parallel fibers separated by a distance d and (**b**) color-coded magnetic field vectors due to the current carrying neighboring fibers.