While experimenting with different node setups, I had the need for a couple of antennas. The information I found on the internet for 868 MHz antennas did not prove accurate, so for completeness I include the calculations for antenna lengths for LoRa applications in the 433 and 868 MHz band.

An antenna is a conductor in the form of a spline, electrically connected to the communications module through a transmission line. The antenna diameter hardly has any importance, as long as the antenna stays in the spline form. The most effective antenna has the same length as the length of the wave it is used for. For practical purposes, half or a quarter of that length will suffice. Most LoRa antennas are a 1/4 wavelength.

The wavelength of a frequency is calculated asv / f, where v is the speed of the transmission and f is the (average) transmission frequency. In air, v is equal to c, the speed of light, which is 299.792.458 m/s. The wavelength for the 868 MHz band is thus 299.792.458 / 868.000.000 = 34,54 cm. Half of this is 17,27 cm and a quarter is 8,63 cm.

For the 433 MHz band the wavelength is 299.792.458 / 433.000.000 = 69,24 cm. Half of this is 34,62 cm and a quarter is 17,31 cm.

A piece of wire of 8,6 cm therefore will do for a LoRa application in the 868 MHz band. The exact length is a major factor in the quality of an antenna. Unless the antenna is directly soldered to the LoRa module, use 50 ohm cable and certified connectors for any transmission line.

A dipole is really good and simple and a simple ground plane even better. Simple dipole has your radiating element in one direction and a ground wire in the opposite direction. If you want a bit more range, 5/8 wave will slightly squash the radiation envelope.

Did you see this?

From the sites I’d found, for 915MHz 7.8cm was the desired radiating antenna length. For 902.3MHz single channel devices I got 7.9cm.

for full wavelength: 299.792.458 / 915.000.000 = 32,76 cm

for half wavelength: 32,76 cm /2 = 16.38 cm
3( for quarter wavelength: 16.38 cm /2 = 8.19 cm
Is this correct?

Yes, that is correct.

As for the cable, will any diameter do?

Any 50 ohm cable will do. You choose the diameter according to the physical circumstances, e.g. thin cable inside an enclosure, thicker cable for mounting indoors. Different cable have different cable loss, so for longer ranges you might want to check into that.

Unless the radiating frequency for 915MHz is way off, 7.9cm is not the correct aerial length. It should be about 8.2cm (see calculations and reference for lambda calculations above).

The 7.9cm reference was for 902.4MHz. I will have to look at why a number of calculators gave me 7.8cm for 915MHz and why your calculation gives 8.2cm.

Well, that part is easy. Follow this link for the theory. Note that under ‘normal’ circumstances, the transmission speed v is ‘the speed of light’, c. Also note, that this is NOT 300.000.000 m/s, as most calculators use, but rather 299.792.458 m/s. The difference for our calculations is substantial. For instance, its the difference between an antenna of 7.8cm and one of 8.2cm.

A practical problem that I have here is: With SX1276MB1MAS I can reach up to 500 m with 868 MHz with the long antenna. However, for 433 MHz with the small antenna, I just reach up to 20 m which is so short range.

In the past, for a particular project, I spent a lot of time checking and tuning antennas on a typical ISM device, the RFM22. My conclusion after around a year of testing is that the theoretical calculated length, whilst a good starting point, is very rarely the optimum length for maximum radiated signal.

There is often an assumption that the output impedance on typical ISM modules is always exactly 50ohm. Whilst this is a reasonable assumption for a high end communications device, typical ISM modules use low cost components, and probably not high tolerance ones. It would seem likley therefore that the output impedance of an ISM device will vary from 50ohm.

Typical improvements in radiated power I was seeing by tuning the antennas for each individual ISM module were 3dB and in some cases as much as 6dB.

By all means calculate the antenna length exactly, but I would be surprised if that is always going to be the ‘best’ length.

We have the same observation. If you hook up a quarter wavelength ‘simple wire’ antenna to an analyser to optimise it for VSWR you usually see that the actual length for optimal VSWR is significantly shorter than the theoretical quarter wavelength.

That’s because a simple quarter wave should work as a dipole and needs a quarter wave counter pole.
Normal the counter pole will be the GND plane of the PCB. Or when you hook up a quarter wave to a analyzer (most of the time you do that with a piece of coax between it). You trim the quarter wave to the that.

@lex_ph2lb nope it’s not the ground plane. We measure and trim our antenna’s always with a ground plane. The reason that a quarter wavelength antenna is shorter than the theoretical wavelength is the velocity factor, or antenna shortening factor.
The theoretical wavelength is calculated in free space, with the electromagnetic waves travelling at the speed of light. In any other medium, like copper wire, they move slower, hence the quarter wavelength in the wire is shorter than in free space, requiring a somewhat shorter antenna.
The order of magnitude of the delta in antenna length is around 5% so indeed noticeable!

Is there a simple formula for calculating the exact antenna length taking into account the velocity factor, or will the 5% rule of thumb be sufficiently reliable in practice?

868 MHz band LoRaWAN frequencies range from 867.1 MHz to 868.5 MHz (FSK not included) or 867.1 MHz to 868.8 MHz (FSK included) with corresponding middle frequencies 867.8 Mhz and 867.95 MHz respectively.
The optimal LoRaWAN antenna length for 868 MHz band should thus be calculated with (velocity factor not included):

299,792,458 / 867,800,000 = 0.345462 m (FSK channel not included), or
299,792,458 / 867,950,000 = 0.345402 m (FSK channel included)

In practice this will make too little difference (with using 868.0 MHz for the calculation) to be significant.

For LoRa modules, then I would suggest the answer is no.

There will be too much variation in the components that match the actual output of the SX127x (LoRa device) to the physical antenna to make an exact calculation possible, unless you know exactly what the matching is.

I am not sure that this should be a surprise, even with top end radio gear antennas should be ‘tuned’ using SWR meters to ensure a good match. If it was possible to calculate the exact length of an antenna then there ought to be no need for antenna tuning at all.