A lossless line having an air dielectric has a characteristic impedance of \(400 \Omega\). The line is operating at \(200 \mathrm{MHz}\) and \(Z_{\text {in }}=200-j 200 \Omega\). Use analytic methods or the Smith chart (or both) to find \((a) s ;(b) Z_{L}\), if the line is \(1 \mathrm{~m}\) long; \((c)\) the distance from the load to the nearest voltage maximum.

z_in = 0.5 - j0.5. #tag_title# Step 2: Calculate the reflection coefficient (s) #tag_content# The reflection coefficient can be calculated using the following formula: $$ s = \frac{z_{in} - 1}{z_{in} + 1} $$ By substituting the calculated z_in, we can find the reflection coefficient s: $$ s = \frac{(0.5 - j0.5) - 1}{(0.5 - j0.5) + 1} $$ #tag_title# Step 3: Determine the electrical length of the transmission line (βl) #tag_content# We can find the electrical length of the line (βl) using the given line length and the frequency: $$ \beta l = \frac{2\pi}{\lambda} l $$ Where λ is the wavelength, which can be calculated as: $$ \lambda = \frac{c}{f} = \frac{3 \times 10^8 m/s}{200 \times 10^6 Hz} $$ #tag_title# Step 4: Calculate the load impedance (Z_L) #tag_content# The load impedance can be found using the following formula: $$ Z_L = Z_0 \frac{1 + s e^{-j2 \beta l}}{1 - s e^{-j2 \beta l}} $$ By substituting the values for s, βl, and Z_0, we can find the load impedance Z_L. #tag_title# Step 5: Determine the distance to the nearest voltage maximum #tag_content# We can determine the distance from the load to the nearest voltage maximum as: $$ d = \frac{(2n + 1)\lambda}{4} - l $$ Where n is an integer. We can start by taking n = 0 and calculate the distance d. If the result is negative, we can increase n and repeat until a positive distance is found.