# Antenna input impedance calculation

I am attempting to calculate an antenna's input impedance. I have been given this information:

A 50 Ω coaxial line is feeding the antenna a 150 MHz frequency. It has a dielectric constant of 2.25, with an outer conductor of 40 mm. The antenna and coaxial cable are not matched, therefore have a standing wave ratio of 2.7. The first voltage minimum occurs 0.42 m from the antenna.

This is what I have so far: $$\lambda = \frac{c}{f\sqrt{\epsilon_r}}=\frac{3\cdot 10^8}{150 \cdot 10^6\cdot\sqrt{2.25}}\approx 1.33\mathrm{~m}$$

$$Z_{in} = Z_{0} \frac{VSWR +1}{VSWR -1}e^{i \frac{2 \pi}{\lambda}}$$

$$Z_{in} = 50 \frac{2.7 +1}{2.7 -1}e^{i \frac{2 \pi}{1.33}}$$

$$Z_{in} = 1.285 - 108.815i$$

It just seems too straightforward.

• So i went a head and had another crack, using the first voltage minimum. Still a bit confused though. $$\beta = \frac{2\pi}{\theta} = 2\pi \frac{3}{4} = \frac{3}{2} \pi \approx 4.412rad/m.$$ $$\theta r = 2\beta d_{min(n)}-(2n-1)\pi rad = \frac{(2*3)}{2}*0.42-\pi = \frac{13}{50} \pi \approx 46.8$$ $$|\Gamma| = \frac{S-1}{S+1} = \frac{2.7-1}{2.7+1} = \frac{1.7}{3.7} = \frac{17}{37} \approx 0.459$$ $$Z_{in} = Z_0*\frac{(1 + |\Gamma|)}{(1 - |\Gamma|)}$$ $$Z_{in} = Z_0 * \frac{(1 + 0.459e^{j46.8})}{(1 - 0.459e^{j46.8})} = 7.767 + j57.54.$$ Commented Apr 2, 2023 at 1:10