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How does a quarter-wave dipole operate?
If a half-wave antenna is the shortest resonant length of an antenna, then how can a quarter-wave antenna function effectively?

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Quarter-wave monopole antennas actually use a ground plane to ‘complete’ the other half of the antenna - so it still operates as if it were a half-wave resonant antenna. This means that a quarter-wave antenna should occupy a smaller footprint on a circuit board. It’s for this reason that they’ve become the most popular form of embedded antenna.

If you take a 916MHz application, a full-wave antenna would need to be just shy of 330mm long. A quarter-wave version would only need to be 87.2mm long. Surface mounted antenna manufacturers can coil this up across multiple layers, but they still require a ground plane of this length to operate as the other radiator. Without a ground plane of one quarter of the wavelength, the antennas efficiency would be compromised and to an even greater degree when operating at lower frequencies.
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This question is answered by attending lectures at the university or by reading professional books. If you ask Google about it you will also get an answer.
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How large of ground plane (mm2) is needed?   Is there a formula based on frequency?

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Will using the ground plane cause any problems with radiated emissions?

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