Impedance Matching: Minimum Reflection vs Maximum Power Transfer
Reflections occur or are noticeable when there is a transmission line involved and that transmission line is long enough for significant reflections to occur. This is generally accepted to be a length of about one-tenth of a wavelength. So, at 1 MHz, the wavelength is 300 metres and so unmatched transmission line problems start at about 30 metres. Higher frequencies naturally have unmatched problems on shorter line lengths.

However, the impedance of a transmission line for radio frequencies of about 1 MHz and above can be taken to be purely resistive. In other words it doesn't present a complex impedance hence it should be matched with an equivalent resistance to avoid reflection problems and this also ties in with the maximum power transfer. So no real problems here.

For an antenna, it can have a highly capacitive impedance if it is regarded as "short". An example being a monopole that is less than one-quarter of a wavelength. The radiation resistance it would naturally present when a quarter wave long would fall from 37 ohms to a much smaller figure when the antenna is shortened. The effective series capacitive reactance rises from near-zero at a quarter wave to tens, hundreds or thousands of ohms as the antenna shortens.

So this is an example of where using an inductor (a conjugate component) can cancel the short antenna's capacitive impedance and allow a better transfer of power.


  An antenna is used to match a circuit impedance (50 Ohm) to the free
  space impedance (120 pi). Ideally we get no reflection, thus I would
  expect that all the power has been transferred to the environment


Of course there is a reflection - that is the mechanism by which we get an impedance transformation to that of free space at a particular frequency. And, adding a conjugate component to cancel out the inherent capacitive reactance of a "short" antenna doesn't alter how the antenna works but it does allow a better transfer of power.