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What are my requirements as a licensed Amateur Radio operator in the US to keep to a single frequency? How much drift can I have during transmission and still be within my license? Does it depend on the mode or band? Note that I'm not asking about the band plan, simply the FCC rules.

I'm asking in relation to making transmitters. I'm sure a crystal (20ppm) is more than sufficient, but should I decide to make a simpler oscillator, how much error is permissable?

If you want to add how much error is preferable for a given activity (DX, contests, rag chewing, etc) then that's fine, but please note that I'm primarily interested in what my license permits, rather than what other operators might prefer (at this time, anyway).

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Your frequency error is the difference between where your transmitter indicates you are transmitting, and where you actually are transmitting.

The FCC does not regulate where you think you are transmitting. They only regulate where you are actually transmitting. Consequently, your frequency error can be whatever you want, as long as you keep it in the bands permitted by your license.

If you are concerned about drift during a transmission, I wouldn't. Anything you build today is almost certainly more stable than tube equipment that was built decades ago. Your problem with drift will be that the other station can't follow you before it is violation of any regulation.

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Part 97 doesn't have anything to say about frequency error or drift in particular. The question pool, however, does:

T1B09 (D) [97.101(a), 97.301(a-e)]
Why should you not set your transmit frequency to be exactly at the edge of an amateur band or sub-band?
A. To allow for calibration error in the transmitter frequency display
B. So that modulation sidebands do not extend beyond the band edge
C. To allow for transmitter frequency drift
D. All of these choices are correct

Of the sections of Part 97 cited, §97.301(a-e) merely defines the permitted bands, but §97.101(a) is:

In all respects not specifically covered by FCC Rules each amateur station must be operated in accordance with good engineering and good amateur practice.

Thus it would seem that in the opinion of the authors of the question pool, “good amateur practice” includes taking precautions to avoid inadvertently transmitting out of band. Having a well-calibrated and low-drift transmitter would merely be another such precaution.

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To add to Phil's answer the only time inaccuracies in frequency are of concern to the FCC are when you're at the edge of the band. As Kevin's answer alludes to, as long as you aren't transmitting out of band, the only person who is concerned about what frequency you're transmitting on is you, and possibly someone you planned to meet on the air.

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