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I'm rather puzzled with MagSafe connector - the close-up photos of the part which is on the device show that there're no holes, only round curved plates and the contact pins on the opposite connector part are just placed against those plates.

In usual connectors there's usually a rather long thin pin that is inserted into a variation of rather long thin hole and this provides huge contact area and also the act of insertion cleans the surfaces and this addresses possible dirt on contacts.

Looks like there's no "insertion" in MagSafe - the connector parts are just placed against each other (and held together by magnets) and the pins are just placed against those curvy plates and I suspect the area through which current actually flows is rather small.

I wonder how would such connector address dirt on contacts and also pass rather high power (up to 85 watts) without overheating, burning and sparking?

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This is like a wiping contact without the wiping ;-).

When you insert an IC into a socket or plug in a USB plug and similar r=the primary action is metal sliding along a surface and coming into face to face contact. There will be some form of "spring" pressure to hold them in contact. Once gross oxide film is removed you usually get OK connections. Rub off all the oxide films and in many cases they would weld together.

In the case of the Magsafe connector it appears that the mating surfaces are Gold - probably Gold plus something to make it wear less. Gold does not form an oxide film (to iner) or perhaps rather, what film does form is extremely tenuous and almost wiped off by mechanical contact.

Old style Magsafe

enter image description here

New style Magsafe (broken)

enter image description here

Note that connectors that appear to mere mortals to work in exactly the same way (magnetic latch, surface contact, pull apart disconnect) have been used for many years on eg deep fryers for safety reasons.

BUT note that the page that you cited says

  • Common complaints include plug separating from the cord, transformer shorting, and pin springs losing elasticity

Without spring pressure it will not make good contact.
With not enough spring pressure it makes not enough contact :-0.

As for how well they work - probably well enough but not as well as a properly designed connector.

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Power is irrelevant, the important parameter is current. At 85W and 16.5V the current is just over 5A. One way the copy with it is having 2 connections for both \$GND\$ and \$V_+\$, so that each pin only has to carry 2.5A. Then the connections are low resistance. If both ends are properly shaped gold-plated connections like these can have a resistance of m\$\Omega\$s, let's say 10m\$\Omega\$. 2.5A \$\times\$ 10m\$\Omega\$ is a voltage drop of \$25mV\$, and a power loss of \$62mW\$. Nothing to worry about.

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