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I have heard that when an antenna is bent, it can reduce the performance of the antenna and thus limit the range of a drone. This is mentioned in this video at 5:29. I am wondering why a bend in an antenna would cause this loss of RF signal? And how significantly would having slightly bent antennas on an XM+ receiver actually impact the range of a drone?

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Antenna design is typically done to tune the transmission to a combination of specific wavelength frequency and polarity. In the case of whip antennas like those usually used on radio receivers, these are monopole antennas tuned to the 2.4ghz band (at least, the FrSky ones are. Some radios transmit on other frequencies).

By bending the antenna, you change it's ability to radiate on the proper frequency. Generally you will still get some energy on the desired frequency, but the signal will be worse because it is less efficient. It's worth noting that this only applies to the active element; in the case of the whip antennas, the black cases portion is simply feeding the short clear cased bit at the end. The only part that matters for signal strength is that bit at the end, assuming you don't damage the wire in the black cased portion.

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Antennas are weird and the answer isn't really clear cut.

As long as the wire within the antenna isn't damaged, it should work fine (1, 2).

Sometimes bends in antenna can improve it, like with NASA's Evolved antenna for the ST5 spacecraft.

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