# Impedance matching for an antenna

I've been working on a project for school which involves designing matching networks for antennas (meant to be driven at 2.4 GHz) being connected to a 50 Ω line, and it led me to consider the following question.

The reflection constant can be defined as $$\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0}$$ Where $$\Z_0\$$ is the characteristic impedance of the transmission line terminated by a load $$\Z_L\$$.

This equation seems to imply that for minimized reflection, $$\Z_L\approx Z_0\$$ is desirable. Since the reflective power losses are determined by $$\\left|\Gamma\right|^2\$$, this seems to imply that for optimal power transfer, $$\Z_0\$$ should be made as close to $$\Z_L\$$ as possible.

However, from the analysis of the lumped element model for circuits, optimal power is transferred to the load when $$\Z_L=Z_0^*\$$.

In the case of the 50 Ω source, there's no issue with these two equations since the line impedance is purely real.

However, if the line impedance wasn't real, what would be the best thing to do?

My guess is that because the frequencies correspond to wavelengths on the order of 10 cm, and most adapters are at least a centimeter, the lumped element model isn't applicable and it would be best to match the impedance to the characteristic impedance of the line rather than the conjugate.

However, one of the textbook's (Balanis) I'm using explicitly mentions that it's important to match antenna impedance to the conjugate of any output impedance from the source, which suggests that the lumped element model is still relevant somehow.

• You might could pickup an amatuer radio handbook. their you could find a bit more digression on that issue. Go for a pre 2000 edition. Jan 1, 2023 at 2:50
• This post might help, they have a similar question. electronics.stackexchange.com/questions/387192/…
– qw0
May 8, 2023 at 5:42