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I have seen many RF PCBs with a thick, rounded area of exposed copper going all around the PCB, like in this example:

Picture of PCB with an exposed copper perimiter

My intuitive thought is that this is for some kind of grounding or shielding can, however I have seen this on PCBs that had none of that.

What is the design reason for this characteristic, rounded strip?

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Think of your PCB as parallel plate waveguide, with the top and bottom ground planes acting as parallel plates.

Now, since you want waves neither exiting nor entering this waveguide, you'd try to build a "wall" around them – or more, a fence, which the small vias around the edge do.

The fact that the ground plane is exposed around the corners (i.e. no solder mask) might point to the PCB potentially being mounted in a conductive, hence shielding if grounded, enclosure.

Rounded corners are nicer to handle, and don't break off. Also, æsthetics.

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Probably every free space of your PCB in grounded to prevent electrical interference as Marcus Müller explain in his answer. However the ESD logo on the PCB

ESD Symbol

suggest that the components are susceptible to electric discharge.

As I can read here: Antistatic discharge method in PCB design

(automatic transation from another language I suppose...)

If the board will not be placed in a metal box or shielding device, the board of the top and bottom of the chassis ground wire solder resist can not be painted, so that they can be used as ESD arc discharge electrode.

So the unpainted corner can be a discharge electrode for user hands.

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