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How do Starfleet comm badges work as far as connections? One user taps their badge and says “me” to “receiving person”.

It’s obviously heard as an inter-comm on the ship, so I can understand how that works (sender transmits over ship’s intercom, receiver taps and closes the connection, privatizing the conversation).

But what about off the ship? How do comm badges work then? There would be no communications array to manage badge connections like there would be onboard the ship. So how do away teams’ comm badges connect?

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Shipboard

As you surmise, when aboard ship, the commbadge offloads all the routing to the ship's main computer:

During voice operations, the normal procedure involves a crew member stating his or her name, plus the party or ship area being called, in a form that can be understood by the computer for proper routing. Examples: "Dr. Selar, this is the captain," or "Ensign Nelson to Engineering." The artificial intelligence (Al) routines in the main computer listen for intraship calls, perform analyses on the message opening content, attempt to locate the message recipient, and then activate the audio speakers at the recipient's location. Personal communicator -> Short-range RF pickup -> local subprocessor -> optical data network -> main computer core.

Planetside

The range of the communicator is very limited

due to the small size of the STA [subspace transciever assembly] clear voice signals will only propagate 500 kilometers. This is a tiny fraction of the 40,000 km required to contact an orbiting spacecraft, so it is the spacecraft that must become the active partner in order to receive the communicator's lower-power signals, and transmit correspondingly high power signals to the communicator's receiver. Audio pickup -> voice discrimination -> data encryption -> control program -> local transponder / subspace receiver

Communications between the starship and a destination typically 38,000 km to 60,000 km away from the antennae are handled by the [hull-mounted RF and subspace] radio systems. Situations encountered cover a broad range, but most notably include discussions with planetside governments, communication and control of Away Team operations ...

Given this, it seems that likely the comm badge still routes its traffic via the ship's main computer systems (when available) through its STA (which makes sense, as the ship keeps as constant a lock on the comm badges as possible for transport, and would allow for monitoring and logging of away team activity while planetside). When it is not possible to communicate with the ship, the control program within the badge itself is presumably capable of cutting out the middle man and establishing communication directly either through the RF transponder (thus acting effectively like a cell-phone) or the STA, and the control program is responsible for identifying whether a signal it is receiving is intended for this unit or not.

(As a side note that the technical manual claims "the communicator is a line-of-sight device away missions", though this seems to have been contradicted plenty of times on screen, so I am disregarding that paragraph.)

All images and quotations taken from the TNG Technical Manual.

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    If I recall correctly, there's a passage that implies communicators really do nothing onboard the ship - the ship is constantly listening, and it uses pickups in each room. The badge-tap is accepted as habit, and the system makes the appropriate noises, but it doesn't actually do anything.
    – T.J.L.
    Commented Aug 6, 2018 at 13:12
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    "Control of the personal communicator while aboard a Starfleet vessel is a matter of preference and habit. To initiate a voice call, it is simply a matter of tapping the front of the badge to confirm to the STA that the message is meant to go out. This may seem redundant, as the intrashipcomm system is constantly monitoring and routing voice transmissions, but it is a good practice to learn. During away team operations, the tap is essential to preserving internal battery power. The tap activates a dermal sensor to relay a power up command to the STA."
    – Ty Hayes
    Commented Aug 6, 2018 at 13:45
  • That's the one.
    – T.J.L.
    Commented Aug 6, 2018 at 14:14
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    I truly wonder if is there a question that the TNG: Technical Manual can't answer.
    – Sekhemty
    Commented Aug 6, 2018 at 16:47
  • Ahhh... technobabble: "the spacecraft that must become the active partner in order to receive the communicator's lower-power signals, and transmit correspondingly high power signals to the communicator's receiver" Great. You're on an away mission and can hear from the Enterprise because they're broadcasting at 40,000 watts. Your combadge still only broadcasts at 0.5 watts and can't reach back out. Just like a cell tower could broadcast at high power so your phone can receive it, but your phone is limited in its power, so it can't respond.
    – FreeMan
    Commented Feb 26 at 15:58

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