# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 3 Electrical Modeling of CNT Networks
### 3D Models for CNT Networks Simulations
#### 3.4.2 Resistive-Capacitive 3D Networks  
When the CNT network is formed within an insulating polymeric matrix, from an electrical point of view the overall impedance must include not only the resistive and capacitive contribution of the conductive paths, but also the contribution of the hosting polymer. For this reason, the equivalent electric circuit-based models are usually improved by including parameters that account for the polymer properties affecting the nanocomposite electric behavior.  
The computation of the resistive and capacitive contributions to the reinforced epoxy frequency-dependent electrical impedance due to the CNT distribution in the polymer is addressed in [50; 51], where the resistive model presented in [52] is improved to account for the capacitive effects between neighboring nanotubes. In the equivalent circuit, the base resin is represented by a resistance and a capacitance in parallel, respectively given by  
\[R_{res}=\frac{L_{i}}{\sigma_{res}\big{(}L_{j}L_{k}\big{)}}\;with\;i=x,y,z\;and \;j\
eq k\
eq i \tag{15}\]  
and  
\[C_{res}=\epsilon_{0}\epsilon_{r}\frac{L_{j}L_{k}}{L_{i}}\;with\;i=x,y,z\;and\;j \
eq k\
eq i \tag{16}\]  
in which \(\sigma_{res}\) is the resin dc conductivity, \(\epsilon_{0}\) is the vacuum dielectric permittivity, \(\epsilon_{r}\) is the resin relative dielectric permittivity, \(L_{i}\), \(L_{j}\), \(L_{k}\) are the fixed volume dimensions along the three axes. The capacitance between two neighboring nanotubes is computed by consid  
Figure 5: Simulated and measured resistance of a 3 \(\times\) 3 mm\({}^{2}\) spray coated CNT network. Data extracted from [49]. Simulation results are obtained by means of the resistive 3D network model described in the cited work.  
ering an effective parallel-plate geometry for the nanotube-insulator-nanotube system, relying on a simplified equivalent topology, and therefore approximated by the capacitance of a parallel-plate capacitor \(C_{gap}\), in which the armors are the tubes' surfaces, and the dielectric is the polymer:  
\[C_{gap}=\frac{\epsilon_{0}\epsilon_{F}S_{CNT}}{d} \tag{17}\]  
where \(S_{CNT}\) is CNT cross-sectional area, and \(d\) is the distance between the two CNTs.  
Each percolation path identified across two sides of the simulated box volume introduces a parallel branch, accounting for the material changing its behavior from insulating to conducting. Each branch includes a series of a resistor and an RC parallel. The series resistance is the CNT resistance \(R_{nm}\) given by Equation (11); the RC parallel includes the tunneling resistance, \(R_{tunnel}\), of Equation (12) and the capacitance originating from the gap between two neighbor CNTs filled with the dielectric polymer, \(C_{gap}\), of Equation (17). The schematic view of the equivalent electrical 3D network is shown in Figure 6. The equivalent electric impedance can be used to compute the overall frequency-dependent equivalent electrical conductivity by knowing the investigated volume dimensions along the three axes. The authors confirmed the consistency of the simulation results by comparing the simulated electrical percolation threshold with that reported in several other works in the literature.  
In [53], a lumped-parameters equivalent electric circuit is proposed to model the ac electrical input impedance of CNT-epoxy composites. The resistances and capacitances of a simple electrical network are extracted by fitting the measured frequency-dependent electrical impedance of composite samples realized with different thicknesses, in a frequency range spanning from 100 Hz to 1 MHz. The equivalent circuits are composed of an RC parallel branch, representing the overall material behavior, and the smallest number of RC series, all electrically connected in parallel, needed to fit the measurement. Although such an approach represents an easy tool for including an already existent device in a circuital simulator, it cannot be used to design a material fabricated by using a different recipe, or the same material in a different frequency range.