# Knitted Microwave Transmission Line for Wearable Electronics
## 2 Materials and Methods  
### Microstrip Transmission Line Model  
Microwave transmission lines enable the transmission of high-frequency signals from a source to the load. In the case of wearable wireless systems, the signal source is the transmitter output and the load is the textile antenna, as presented in Figure 1. Each of these elements is characterized by its impedance, as shown for the case of the microstrip line in Figure 2, where the source of the signal \(a_{1}\) is connected to port 1 of the line, while the output signal \(b_{2}\) at port 2 is connected to the load (textile antenna, potentially). To avoid reflection losses, the transmitter output impedance \(Z_{g}\) should be equal to the characteristic impedance of the line \(Z_{s}\), and this should be equal to the antenna input impedance \(Z_{l}\). Typically, high-frequency elements have an impedance aimed at the reference value \(Z_{0}=50\)\(\Omega\), and any component of the system, such as the transmission line, should have a very similar value of impedance. In high-frequency systems, where unbalanced transmission lines (such as the microstrip line discussed herein) are predominantly used, the characteristic impedance is standardized to a value previously common in coaxial lines. Except for the 75 \(\Omega\) cable used in television systems, the majority of coaxial cables and connections in everyday use have a 50 \(\Omega\) characteristic impedance. These decisions are justified by the fact that an air-filled coaxial line has maximum power capacity at a characteristic impedance of roughly 30 \(\Omega\) and minimum attenuation at a characteristic impedance of roughly 77 \(\Omega\). Therefore, a 50 \(\Omega\) characteristic impedance is a trade-off between highest power capacity and least attenuation [40]. The measure of impedance matching that is used in this paper is the \(S_{11}\) element of the scattering matrix, defined in logarithmic scale (in dB) as follows (1) [41]:  
\[|S_{11}|=20\cdot log_{10}\bigg{(}\frac{Z_{s}-Z_{0}}{Z_{s}+Z_{0}}\bigg{)}, \tag{1}\]  
where  
\(Z_{s}\) is line impedance;  
\(Z_{0}\) is the reference impedance of the system (50 \(\Omega\)). In the case where the impedance of the elements is equal to the reference impedance \(Z_{0}\), there is no signal reflection and the value of the \(S_{11}\) module approaches minus infinity. The greater the impedance mismatch, the greater the signal reflection and the value of the \(S_{11}\) module. In extreme cases (short circuit or open circuit), the value of this parameter is 0. It is desired then for the transmission line to have as small an \(S_{11}\) module as possible; however, practically, values below \(-10\) dB are acceptable, since this corresponds to the case when the reflected signal power is 10 times smaller than the power of the inserted signal. Another important parameter describing the transmission line is signal attenuation. Referring to Figure 2, it is the ratio of the output signal \(b_{2}\) at port 2 to the input signal \(a_{1}\) at port 1. It can be described by the parameter \(S_{21}\) of the scattering matrix, which is the transmittance from port 1 to port 2, and is defined in logarithmic measure as follows (2) [41]:  
\[|S_{21}|=20\cdot log_{10}\bigg{(}\frac{b_{2}}{a_{1}}\bigg{)}, \tag{2}\]  
where  
\(a_{1}\) is the signal inserted at port 1;  
Figure 2: Microstrip transmission line impedances and signal definition. \(b_{2}\) is the signal leaving port 2. The value of the \(S_{21}\) parameter should be as high as possible, ideally being 0 for a lossless line, for which \(b_{2}=a_{1}\). In practice, lines with attenuation not exceeding a few decibels are acceptable. The microstrip line, which has been extensively studied, is a widely used form of planar transmission line due to its easy miniaturization and integration with passive and active microwave devices, as well as its ability to be made using photolithographic methods [40]. Figure 3 illustrates a microstrip line's geometry.