It’s common when giving tech support to ask someone to “unplug, wait, and then plug back in”. This is because many people toggle the wrong switch or do something unexpected. Asking to unplug, wait, and then plug back in is a shortcut instruction to dummy proof and make sure the device gets all the way turned off and then back on for a reset.
Coming from a person that kept telling people to pull the plug on the router I can answer a bit.
There is always a way to transform energy from the wall socket into something usable, like from 230V AC to 12V DC.
This is usually done by a) a transformer ( no longer used ) and b) a step-down converter.
Now the issue with the step down converter is that it uses a dedicated pwm controller ( a processor ) that needs a sort of well regulated voltage to work. However as the wall adapter housing the step-down converter is all inside the “brick” it gets hot, and as it gets hot the capacitors that are used to regulate the Voltage for itself and for your appliance decrease in capacity and the Voltage starts oscillating to much, causing either a) the pwm controller to go haywire, or b) making your router’s processor behave unexpectedly.
As an example, a typical oscillation with a good capacitor might be something like +/- 0.02V, but with a bad capacitor it might go to 0.25 or more, and this causes weird things to happen.
By powering down the wall brick, all of the power inside the processors go to Zero allowing them to start relatively fresh, even thou later it might repeat as the capacitors and other components get hot.
Note that capacitors also go bad as they age regardless of temperature ( temperature accelerates this process ), and there are also “bad” capacitors on the market.
In addition to what others have said, waiting a minute before plugging it in allows any capacitors to discharge. Juts turning it off and immediately back on risks some part of the circuit staying energised and not rebooting properly, because a charged capacitor can maintain some voltage for a while.
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