## 2 Materials and Methods
### Electrical Conductivity Characterization of CFRPs
#### 2.1.2 Transverse Conductivity
As depicted in Figure 7a, the contact impedance can be translated into an equivalent circuit comprising a resistor-capacitor parallel circuit pair. The magnitude of this impedance \(|Z_{c}|\) is expressed below:  
\[|Z_{c}|=\frac{1}{\sqrt{\frac{1}{R_{c}^{2}}+\frac{1}{X_{c}^{2}}}} \tag{10}\]  
where \(R_{c}\) and \(X_{c}\) are the resistance and capacitive reactance of the matrix region at the contact. For a curved matrix geometry, as illustrated in Figure 7b, the contact resistance and capacitive reactance is estimated by discretizing the matrix region and integrating. Their simplified expressions are as follows:  
\[R_{c}=\frac{\rho_{m}}{len_{f}I} \tag{11}\]  
\[X_{c}=\frac{1}{2\pi fC_{c}} \tag{12}\]  
The contact capacitance \(C_{c}\) can be further expressed in terms of the material's relative permittivity \(e_{r}\) and permittivity of free space \(e_{o}\) as follows:\[C_{c}=\varepsilon_{o}\varepsilon_{r}l_{c}I \tag{13}\]  
The term \(l_{c}\) denotes the fiber contact length, which is defined as the total length of the segment along the fiber direction where fibers consistently remain in close proximity. It's important to distinguish this term from the sample length along the fiber direction, as fibers exhibit a wavy, nonlinear nature, as illustrated in Figure 5.0 \(l_{c}\) was chosen as a percentage of the overall RVE length for which the wavy fibers maintain proximity. In Equations (11) and (13), the term \(I\) is a constant and can be expressed as shown below:  
Figure 5.1  
Figure 5.2 Transverse conductivity model outline. The energy required to transmit an electron through the matrix barrier is calculated from the energy, current, and voltage relationship as follows:  
Figure 5.3  
where \(I_{rms}\) is the RMS current of the signal, and \(\Delta t\) is the time taken by the particle to pass through the barrier. Upon simplifying Equation (15) further, we obtain the following expression for the barrier potential energy \(U\):  
Figure 5.4  
Figure 5.5 Schematic showing the fiber contact length (\(l_{c}\)) due to fiber waviness. Figure 5.6 Fiber pair schematic with an enlarged view of the contact region showing the wave functions (\(\Psi\)) for different regions along the particle tunneling path. Figure 5.7 (a) Equivalent circuit of a fiber-fiber contact pair and its equivalent contact impedance (\(Z_{c}\)) and (b) contact region discretization for determining effective contact resistance (\(R_{c}\)) and capacitance (\(C_{c}\)). where \(q_{e}\), \(c\), and \(n_{m}\) are the electron charge, speed of light, and refractive index of the matrix, respectively. The term \(c/n_{m}\) denotes the speed of electrons in the matrix medium. The energy of the particle E is expressed as follows:  
Figure 5.8  
The shortest conductive path is analyzed next to determine the effective transverse impedance. Using Equation (9), the specific tunneling probabilities \(\left(P_{link}\right)_{n}\) associated with each \(n^{th}\) fiber-fiber link within the conductive path can be calculated. Figure 9 visually represents the electron tunneling process along the shortest route. The cumulative tunneling probability across the RVE, denoted as \(P_{net}\), can be expressed as the product of individual tunneling probabilities for all links as shown below:  
Figure 5.9  
where \(N_{l}\) corresponds to the total number of fiber-fiber links in the conductive path. In a hypothetical scenario where all the fibers along the conductive path are touching (\(P_{net}=1\)), the overall RVE impedance can be determined by summing the resistance across each fiber cross-section within the conductive path. This cumulative impedance, denoted as \(R_{net}\), can be mathematically represented as follows:  
Figure 5.0  
where \(N_{f}\) denotes the total number of fibers in the conductive path and \(\left(R_{ft}\right)_{n}\) is the resistance across the \(n^{th}\) fiber cross-section. Here, the subscript \(ft\) refers to the transverse fiber resistance.