Total posts : 45366
Regarding the Pi matching references you found, the inductance and capacitance will have little to do with the inductance of a loading coil so the values may not be anywhere near 250 uH. There are two things going on here. The pi network functions to provide an acceptable impedance to the transmitter and does so by transforming the impedance at the antenna feedpoint to a suitable value. The second is the loading coil cancels the capacitive reactance at the antenna so the feedpoint impedance is resistive. Unfortunately, at part 15 frequencies and antenna lengths this resistance is a very small number which the pi or other network needs to transform up to a value suitable for the transmitter (usually but not always around 50 ohms).
If you look a few posts back you will see that Rich has modeled a short part 15 antenna and the impedance is Z = .106 – j2523 ohms. The primary way we express a resistive and reactive impedance is in this manner which is known as a complex number.
The resistive term is .106 ohms. The reactive, or imaginary, term is -j2523 ohms. The negative sign means this is a capacitive reactance. This is calculated Xc = 1/(2*pi*frequency*capacitance) ( units are Hz. and Farads ). From Rich’s number and solving for C for 1600 kHz., C = 1/(2*pi*f*Xc) = 1/(2*3.14*1600000*2523) = 39.4 picofarads.
The inductance required to resonate (cancel the Xc) at 1600 kHz. comes from XL= 2*pi*f*L. Solving, L = XL/(2*pi*f), so L = 2523/(2*3.14*1600000) = 251 microhenries.
The equivalent circuit for the antenna is Xc in series with the XL in series with the radiation resistance of the antenna (.106 ohms) in series with the loading coil resistance in series with the ground resistance. In this example, the XC is -2523 which added in series with the XL which is +2523 gives 0 leaving only the .106 resistive term at resonance for the antenna in series with the other resistances I mentioned. Reactances such as XC and XL do not dissipate or radiate power and that is why we use loading coils to cancel these terms.
At resonance, the XL and the XC cancel and the model includes only the resistive terms.
As Rich pointed out, usually the biggest unknown is the ground resistance since this is comprised of many factors. If you build a coil to give around 250 uH. it will resonate the antenna at 1600 kHz. used in this example and you should make the coil resistance as small as practical by using big (#16 or larger) wire. Unfortunately there is an effect called the “skin effect” which increases the effective resistance of a conductor as frequency increases.
What this boils down to is in practice you should build a coil using fairly large wire to cancel the antenna capacitive reactance and bury at least 10 10′ ground radials and mount the transmitter between the coil and the radial ground. It is great to pursue the theory beyond this, but this approach, as reported from many posters here, is about the best you can do for a legal antenna, and going further is probably a lot of effort for diminished return.
Hope this helps.