[LEAPSECS] Of stepping motors and leap seconds
Tom Van Baak
tvb at LeapSecond.com
Thu Feb 7 05:02:56 EST 2019
A leap second related story problem.
After visit to USNO years ago I wanted to make a leap second desk clock as a thank you gift. The idea was to use a standard 32 kHz quartz clock stepper movement   but drive it with a microcontroller such that during a leap second it slows down for a bit. I've hacked such clocks in the past; it's quite easy. 
Now Google smears the leap second over an entire day; which is quite a long smear. I understand the NTP-related motivation for this. My clock was a very short smear. Note that if you have a digital display no smear is necessary since you can drop a :59 or add a :60 with ease. LED or LCD digits display instantly. But if you have a "real" clock, one with analog hands, you can neither display a :60, nor drop a :59, nor can you do anything instantly. The physics of electro-magnetism, motors, and inertia prevents that. Hence the name "smooth" back then, although thanks to Google, I now much prefer smear.
I never finished the project. But here's the math.
a) For a positive leap second one can tick normally through :57 and :58 and :59 and then reduce the clock to half speed for 2 SI seconds. This will result in the clock spending two seconds to get from :59 to :00 at which point it will again be in sync with UTC at the top of the hour. It seemed the simplest way to get an analog clock to mimic a leap second.
b) For a negative leap second one can tick normally through :57 and :58 and then increase the clock to double speed for 1 SI second. This will result in the clock spending one second to get from :58 through :59 to :00 and it will again be in sync with UTC at the top of the hour. Also very simple.
What I liked about these two methods is the symmetry. Both positive and negative leaps are a smear. In one case you 1/2x the speed for 2x the time; in the other you 2x the speed for 1x the time. A negative leap second is more fun to watch since it's 4x the rate of a positive leap second. Either way, the result is the required one second phase change in the analog dial display.
These ideas were inspired by a vintage Junghans DCF77 / WWVB analog clock. During synchronization it enters a rapid tick mode (like a negative leap second) and when low on battery it enters a half tick mode (like a positive leap second).
Now back to Google and its ~24 hour smear. One feature of their approach is they center the smear around midnight. What happens if this is done with a short smear analog clock?
c) For a positive leap second you could start the 1/2x clock slowdown at :59.5. It will last for 2 SI seconds and end at :00.5. This is the same as case (a) except it starts half second later in order to be centered across midnight.
But given that these analog clock hands can't rest at half-second marks, this solution feels awkward. So another approach might be a 2/3x clock slowdown starting at :59 that lasts for 3 SI seconds and ends at :01. This avoids the half-second problem and still keeps the leap centered. Other numerical solutions follow the same N/M rate pattern.
d) For a negative leap second one could increase the clock to 2x speed at :59 and end at :01. This is the same as case (b) except it begins 1 second later in order to be centered across midnight rather than end at midnight.
There is still symmetry in cases (c) and (d) but the rate fractions get a bit more complicated.
A further level of complication with Google's next smear is (I think) they want to start 12h before midnight (pre-leap) and end 12h after midnight (post-leap). This suggests the actual number of SI seconds on either side of midnight will differ by 1 which may imply the true center of the smear isn't actually at midnight.
Anyway, if any of you are interested in this sort of thing, let me know. The larger topic is not a quartz desk clock per-se, but investigation into different ways to implement short smears, as one might do in a quartz clock, or a cesium standard, or an operating system that wants to get the leap second over with as quickly as possible instead of as slowly as possible.
I'm wondering how the LEAPSECS crowd feels about long or short smears. One way to measure this is the area under the curve. For a 24 hour smear the |error| ramp goes from 0 to 0.5 to 0 over 24 hours; so that's an integrated error area of 6 hour-seconds. For a short smear it's more like 1 second-second.
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