Electronic – Impedance Matching between Two Antennas

If so, do I place a 50 ohm resistor(50ohm from inductor or capacitor) before the input of the next stage to match,

For matching RF connections, receive or transmit, don't, whatever you do, start sprinkling resistors around thinking you will do good, because they will absorb and lose you more power than you would ever lose through mismatch reflection.

Treat each end of the transmission line independently. Match the antenna to the downlead. Match the downlead to the receiver. Obviously this will be easier if the downlead impedance is equal to at least one of those, and easiest if it equals both.

Given a choice between 50 and 75 cables for a radio downlead, 75ohms is invariably chosen. This is because it's lower loss than 50 ohm cable. It also happens to be a 4:1 impedance ratio with the 300 ohms of a folded dipole, which means an easy-to-build 2:1 balun will do the job. 300 ohms twin wire is a good choice too, if the antenna and receiver are 300 ohms.

What's the input impedance of your PCB filter? If it's intended for FM reception and it's single ended, it's almost certainly 75 ohms, if balanced it's 300. Many FM receivers have a built-in balun, so you can connect 75 coax or 300 ohm twin wire downleads.

Could you skip the filter and connect it to the next stage? What's the impedance of the next stage? Most filters are designed between equal impedances, but some are designed to match impedances from input to output over a small range of frequencies. What's yours doing?

If you don't have an FM receiver, but instead some general purpose test gear, it's more likely to have a 50 ohm input impedance. In this case, 50 and 75 are so close that you will lose little signal by connecting them directly, probably less than if you used a 50:75 ohm transformer, which is only 1.22:1 in voltage terms.

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.