Impedance matching for an antenna

Question

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.

Answer 1

The complex conjugate load is where max. power transfer takes place from the source, no matter if lumped elements or not.

Instead of an antenna, you could also connect a load circuit like e.g. a resistor and capacitor (or inductor) to realize max. power transfer from the source into this (conjugate complex) load circuit.

Also note that if you have a transmission line like e.g. a long line feed to the antenna, the complex reflection coefficient is not a constant but will be transformed along the transmission line, so that e.g. an inductive load appears capacitive after lambda/4.

Answer 2

If the source impedance has non-zero reactance then the load impedance can cancel that out for maximum power transfer. But, in practice, the source impedance should be tuned to have zero reactance at the frequency of interest. Sure, nothing is perfect but you can get close enough that it doesn't matter.