Short-Term Repetitive Watch
Error Due to Stepping

by Walt Arnstein

The Accuracy Concept

The subject of watch accuracy is a popular topic among watch fanciers and the discussions that frequently pit proponents of one brand or type of watch against those of another. Quartz watches, some say, are more accurate than mechanical ones, clocks more accurate than watches, fast-beat 8 bps mechanicals are more accurate than their 5 bps brethren, and various permutations of these comparisons are cited.

Yet we rarely see a rigorous definition of just what is meant by accuracy. We have all heard the old proposition, often used in a tongue-in-cheek spirit, that a watch with trompe-l’oeil hands painted on the dial is very accurate because it shows the exact time twice a day, while the almost perfectly tuned mechanical watch may show the right time once in its working lifetime — when it is set. Due to its admirably slow drift from the time standard, it may turn to a pile of rust long before it manages to accumulate a 12 hour error.

The painted-on hands concept is not outlandish. The only aspect of it that makes it seem so is the interval between correct indications. 12 hours simply isn’t acceptable in our hectic existence. Well, how about a clock — one with a mechanism this time — that steps by an hour at a time and shows the correct time exactly on the half-hour, say? Probably still much too rough to be useful. All right, consider then a clock that steps once per minute. Now it begins to sound plausible, doesn’t it? After all, some regulator clocks in schools and train stations during the ’30s and ’40s stepped at that rate and everyone seemed content with the approximation.

Continuing with this line of reasoning, we note that quartz dress watches without second hands often advance in 1/3 minute steps every 20 seconds. Most owners of this type of watch are quite happy with this level of resolution and quickly lose awareness of it.

Finally we have the second-hand-equipped watches, like the quartz watches that advance in 1-second steps and the smoother-running — but still stepping — mechanical watches that advance at 4, 5, 6, 8, or 10 steps per second. Many fanciers of mechanical watches point to the ‘smoothness of motion’ of these watches’ second hands.

As a point of fact, almost all clocks and watches we use today are of the type we have just been discussing. That is, their displays don’t move smoothly; they step and as a result, all are capable of showing the correct time only at discrete moments consistent with their stepping rate. The rest of the time they incur a repetitive error that is a consequence of the stepping nature of their motion. Technically, this type of error falls into the category of quantization error.

Conclusion? All watches and clocks — except clocks that use synchronous motors running continuously off an AC electrical signal — act as if their hands were painted on the dial for a large part of the time. Only at relatively infinitesimal periods of time, they jump from one “painted-on” location to the next.

At this point, we can begin to appreciate the problem of defining accuracy and consider the error sources that degrade it. The long-term accuracy, which is characterized by variations whose effect becomes noticeable only after hours, days, or even weeks; and the short-term-accuracy, which is dominated primarily by the stepping, or quantization error.

This discussion will concentrate mostly on the stepping error.

Characterization of Stepping Error

We all have a gut-level feel for what amount of stepping error we are comfortable with and what we consider excessive. Millions of people accept as a fact of life the relatively large and infrequent 1-second advances of the quartz movement. But how useful is this 1-second resolution when you have to time events to the tenth of a second, say? Obviously, your reading of the time on the dial maybe off by as much as a second. Unfortunately, considerations of power dissipation and battery life make a higher frequency for quartz watches impractical at this time. In the meanwhile, we must sadly conclude that a 1-second-stepping quartz watch isn’t accurate enough to time events or phenomena requiring 1/5 or 1/10 second resolution.

Let’s consider the task of characterizing short-term error. For simplicity, let’s start with the quartz movement, mainly because of its low stepping rate. Assume that the oscillator has been carefully adjusted relative to a time standard in such a way that its long term accuracy is near-perfect. Our best setting for minimizing errors of all types would be to set the watch so that it is 1/2 second fast at the moment the second hand first lands on a given second mark. (This precise timing is almost certainly not possible with today’s quartz watches, but let’s stipulate that we can somehow do it). The hand will now stay there for a full second, (as if painted on the dial), gradually causing the watch to lose time. One-half second into the stationary interval, the watch will be showing the exact time. Another half second later, its indication will be 1/2 second slow. Just then, the second hand will jump forward by a full second, coming to rest on the next second mark and it will now be 1/2 second fast again. This cycle will go on indefinitely. For the interval between time 0 and time 1 second, an expression for the error of the indication as a function of time is given by the equation:

where frac(t) is the fractional part of the time t. The function looks like an endless sawtooth wave, consisting of vertical 1-second-high edges alternating with downward sloping lines connecting.

Quantifying the Repetitive Error

To quantify the above error in a meaningful way, we will need a steady-state measure that is insensitive to the direction of the instantaneous error and concentrates solely on its perceptible magnitude. It must also be an apt measure of the “nuisance value” of the repetitive error. The best measure of this nature already exists in the form of the RMS (Root Mean Square) error function, which is obtained by squaring the instantaneous error’s amplitude and then calculating the square root of its mean level away from zero over time. Note that initially squaring the error magnitude in the process of calculating RMS eliminates all negative values, replacing them with squares, which are always positive.

All of us are familiar with the RMS concept from our household AC power specifications. The 110 Volts, for example, specified for the US home electrical supply is in fact the RMS value of the voltage evaluated as described above. (In actuality, the instantaneous value of the household 110VRMS voltage sinusoid cycles between -156V and +156V).

Applied to electrical power, the RMS voltage of an AC power signal is the signals heating ability. In a similar way, we can think of the RMS value of the stepping error signal in our watch as the “nuisance value” of an equivalent steady-state offset.

RMS error for the various watch stepping rates is easily calculated using calculus for the simple waveshapes above. It is defined by the expression,

Shown below are the RMS errors for most of the available watch stepping rates used today:

Stepping rate,   RMS error,
steps/second seconds
1 .2887
4 .0722
5 .0577
6 .0481
8 .0361
10 .0289

The significance of these figures is easy to appreciate when one considers their intuitive meaning. If you are wearing a watch of any of the above type, and consult it repeatedly at randomly chosen moments, the above figure is the most likely error you will be reading. Keep in mind that the value is independent of the error polarity.

So which kind of watch is the most accurate? If you are timing an event of short duration, for which long-term drift of the movement is not significant, the 10 bps watch will be 10 times as “accurate” as the quartz watch that steps only once per second. That is, consulting your watch at a random time your most likely error will be 1/10 as great as what you are likely to encounter with the quartz — not to mention that the maximum error you can observe will also be only 1/20 second in magnitude, compared to the quartz’s 1/2 second.

The above tabulation puts a different perspective on the issue of super-accuracy in quartz watches. It is all well and good that several of them have long-term accuracy amounting to an error of less than 10 seconds per year. But on a short-term basis, their precision is somewhat tarnished. They are no better than any other quartz watch and much less precise than most mechanical watches, particularly the high-beat models.