An AT&T 2Wire digital subscriber line modem and wireless router uses all solid-state devices that should not make any sound under any normal conditions. If yours is making a clicking sound, it may have an internal short that is making an arc. You should unplug the power and find a repair facility.
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AT&T 2Wire Router
Although the Pace company makes several versions of 2Wire routers for AT&T and other Internet service providers, none of them have relays to make connections that would click under normal operating conditions. The most commonly used 2Wire routers provide four Ethernet ports for wired connections to computers and an internal wireless antenna to make additional connections in a home network. The lights on the front of the units indicate the status of the network signals. Some blink but shouldn't make a sound.
The power light on a 2Wire router blinks green while the system is performing a self test. When completed, it should be solid green. It blinks orange while undergoing a software upgrade and is solid red if it has failed. The Broadway link or DSL light blinks while the system sets up, but stays green when working properly. The Ethernet, USB, or Wireless lights show green when a connection is made and blink at various speeds while transferring data. If a clicking noise accompanies any blinking light, the unit should be inspected by a qualified technician.
An arc occurs when an electronic component fails and electricity jumps from one point to another and often causes a clicking noise. An arc may not cause a 2Wire router to fail, or the failure may not be noticeable in your particular operating mode. However, it is a clear indication of an electrical problem and should be investigated by a qualified technician. Leaving the router on and operating may cause further damage to the equipment or a fire.
An AT&T 2Wire router normally gets slightly warm but never hot. If it has an electrical problem that causes it to overheat, some components may make a clicking sound as they expand. Feel the unit for excessive heat. A properly operating router will be comfortably warm and you should be able to hold your hand on it for several minutes. But if your hand gets hot, the router has serious problems. Take it to a qualified technician for an inspection.