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I'm planning to make a magnetic loop antenna mainly for receiving LW, MW, and SW radio broadcasting bands, as well as perhaps the 2.2km band and 630m band. (Not all of them with the same antenna, I'mm gone start with LW probably, so I can listen to foreign radio stations).

Now, I want to attach that antenna to the input of an RTL SDR, which has a 50Ω coax input. I can use the RTL SDR in direct sampling mode going from basically 1kHz to around 25MHz, so that's technically no problem, but I'm unsure how to attach the magnetic loop antenna (my first one is just gonna be something similar to this).

I'm also planning to make a ferrite core antenna, and see how that works out. But that's for a later question.

Now, I can use online calculators to get things down like the diameter, the number of turns, and the values for the tuning cap. But I'm unsure about how to attach it to the SDR. I have trouble figuring out the coupling loop which then goes to the 50Ω RG58 coax. So, I'd need a little help and/or ideas with that.

OK, now to clarify further, I'd like to build a loop antenna as described here: AM Loop Antenna Calculator - UMR EMC Lab Formula

So according to that and the build instructions I can make a box AM loop antenna for the sort of frequency range I'd like.

The pic up loop in those designes goes once around the entire loop of the antenna, and has nearly the same diameter. But how do I actually design the pick up loop? I.e. I have to match it to the 50Ω impedance of the coax cable. The build instructions just tell me to get a length of wire, and do a single turn around the perimeter of that antenna, but I don't see how that just matches the 50Ω impedance of my coax.

polemon
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You may be overthinking this. To make a long story short, just connect it and tune the capacitor to resonance and don't worry about the impedance of your pickup loop.

When tuning the radio to antenna system for a transmitter, you have to impedance match the load to minimize reflections because power loss is typically a big deal. Reflections lead to high SWR, and high SWR leads to higher power losses in the coax and heating in both the coax and the radio and all these problems lead to lower transmission efficiency.

In a receiver, the power levels are lower, and the losses are much lower, so this is not a big concern. Typically your amplifier makes up for any weakness in the signal, and you are more concerned about noise from both the amplifier itself and surrounding signals. (re: amplifier, see noise figure.)

In the radio antenna system, you need to match the load to 50 ohms. In a transmitter, the load is the antenna. In a receiver, the load is the radio. Usually the feed line between the antenna and the radio is 50 ohms. So you match the 50 ohm radio to the 50 ohm coax to the (maybe) 50 ohm antenna.

In a receiver, the radio is already 50 ohms, so the matching there is already done. Making the antenna 50 ohms helps a bit, but it helps a lot more if the antenna is resonant on the target frequency to maximize the weak signal it gets.

So, in the end, you connect the coax to the radio, and how you connect the antenna to the coax is not that big a deal for a receiver, especially since you won't be pumping watts of power through that connection.

With a small loop antenna (circumference typically less than 1/10 wavelength) used for either receiving or transmitting, the easiest way to tune it is to listen to your receiver and adjust your tuning capacitor for maximum signal strength. The small loop's bandwidth is so narrow that you don't even have to tune in a signal to check for resonance -- when it is off frequency, you won't even get static; so just tune for maximum static at the desired frequency, and when there is a signal there, you'll get that as well. The reason you don't see a lot of discussion about how to tune your pickup coil is because the coil becomes part of what affects the antenna's impedance; so however you connect it, if you tune for maximum signal, you'll already have the best balance between resonance and impedance.

So my suggestion is to connect the loop any way you can to the coax, and then test it. If you are unsure you selected the best way, try other methods and see if the signal is stronger. More than likely, all methods that work at all will work well enough.

user10489
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You don't necessarily need a coupling loop. Nor do you need to worry too much about matching the antenna to 50 ohms. Firstly, because SWR is not very important for receive performance, and secondly because on a cursory search it seems the RTL-SDR at least when modified for direct sampling doesn't have a 50 ohm input impedance anyway.

The simplest connection method is simply to connect the shield to one end of the loop, and the center conductor to the other end. While this works OK, it will suffer from common-mode current on the feedline. If you are planning to put the antenna outside away from noise sources, or utilize the azimuthal nulls in the loop pattern, the common-mode current may be a problem worth solving.

One solution is a ferrite balun, just the same as would be used for a dipole.

Another solution is a "shielded" loop. The shield isn't so much about shielding anything as it is making a balun. However it's not very feasible to make a shielded loop with more than one turn. You could use a smaller shielded loop as a coupling loop within a larger, multi-turn loop.

Some designs employ a small preamplifier installed directly at the antenna feedpoint. This preamplifier usually has a balanced input so no balun is necessary. These are called "active" antennas. LZ1AQ has some nice designs with schematics and kits available.

I wouldn't overthink the problem. I'd recommend building a prototype and iterating on the design, ideally with two prototypes so you can do A/B comparisons. Especially with electrically small antennas like magnetic loops, you want to optimize for signal to noise ratio, not absolute power.

Phil Frost - W8II
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An alternative impedance matching method for the magnetic loop is a series capacitance between each side of the feedline and each side of the the tuning capacitor. Typically from 15 to 50 pF 6kV doorknobs per side has been found to work on my kilowatt transmitting loops for 20 through 160 meters respectively.

In your receive only case, a pair of small receiving air variables, adjusted for maximum signal, may work. Readjusting the loop's main tuning capacitor will be necessary, as these adjustments interact.