# Knitted Microwave Transmission Line for Wearable Electronics
## 2 Materials and Methods
### Microstrip Transmission Line Model
A thin layer of conducting material, of thickness \(h_{g}\), is placed on the lower surface of a dielectric material of thickness \(h_{d}\) and permittivity \(\varepsilon_{r}\). On the upper surface of the dielectric, there is a thin strip of conducting material of width \(w_{s}\) and thickness \(h_{s}\), while \(L\) is the length of the line. The microstrip transmission line is an unsymmetrical quasi-TEM waveguide that requires the connection of the conducting layer (that is located on the bottom surface of the dielectric) to the ground potential, as presented in Figure 2. It is then compatible with the transceivers, which often have unsymmetric output, and with antennas with unsymmetric feeding; it is especially compatible with microstrip antennas that also have a dielectric backed with a grounded conductor layer [42]. In the literature, there are analytical relations used both for designing and analyzing the microstrip transmission line [41; 42]. The first approach allows the determination of the dimensions of the line, assuming the required characteristic impedance. The second approach assumes knowledge of the structure geometry and material properties. On this basis, the line input impedance and other parameters, such as attenuation or the velocity of signal propagation, are calculated. A significant limitation of this approach is that the analytical formulas are derived assuming that the thickness of the conductive strip is many times smaller than the thickness of the dielectric. It is then possible to ignore the edge effects associated with the complex distribution of the electric field at the boundary of the strip and the dielectric. In the case of the textile transmission line presented here, such an assumption is not met because in knitting technologies it is impossible to obtain very thin conductive layers. Knowing the parameters of this technology, it was assumed that strip thickness would be approximately \(h_{s}=0.5\) mm and substrate thickness would be \(h_{d}=3.5\) mm. For this structure, \(h_{s}\) is \(1/7\) of the substrate thickness, and in such a case the analytical approach cannot be applied. To design a knitted microstrip transmission line that operates in a 2.4 GHz band, computer simulations were applied in the CST Studio Suite program (2022.05 release). Simulations were made in a time domain using the Finite Integration Technique (FIT) [43]. Figure 3: Microstrip transmission line dimensions. This allowed us to include in the simulations the significant thickness of the strip in relation to the dielectric. Figure 4 shows a cross-sectional view of the line model in which a complex structure of knitwear is approximated as the homogenous material. The knitted dielectric substrate is modeled as a lossy dielectric and characterized by a relative permittivity \(\varepsilon_{r}\) and loss tangent (\(\delta\)). The conducting surfaces at the bottom of the dielectric and the strip are modeled as perfect conductors. In this model, hexahedral mesh was used, the size of which was automatically adjusted to the geometry of the model. The shortest edge of the mesh was 0.1 mm (in the region of the ground conductor) and the largest was 2.5 mm (in the free space region). The distribution of the module of electric field intensity in this structure is presented in Figure 5. In this case, the complex nature of this distribution in the area of the side walls of the strip, near the dielectric, is visible. The geometrical parameters of the microstrip line were selected considering the limitations of the available equipment and the knitting technology employed for its fabrication. This necessitated the use of a 3.5 mm thick dielectric substrate. The strip height that could be obtained in the considered technology was equal to 0.5 mm, while its width was chosen to be significantly smaller than the substrate; however, this parameter could not be varied over a wide range.