WatchTech

Measuring Q, the Quality Factor

 

In theoretical horlogerie the quality factor, known as Q, is a highly discussed quantity.  Originally developed in electronics to describe the quality of electrical oscillators, Q is now used in many areas to describe the behaviour of oscillators of many types.  In watch and clockmaking it is used to describe balance wheel/hairspring systems and also pendulums.

Simply put Q is the ratio of the energy stored in the oscillator to the energy lost during each oscillation times 2?.  A perfect oscillator with no losses would have an infinitely large Q.  A good mechanical wrist watch has a Q of about 300, a cheap quartz watch a Q of about 3000 and a good pendulum clock a Q of about 10'000.  The best pendulum clocks running under vacuum to eliminate the friction with air have a Q of about 100'000 and today’s atomic clocks have a Q of over 10'000'000'000.  The rate accuracy possible with that good mechanical wrist watch is about 5 seconds a day, the cheap quartz watch about 1 second a day and the pendulum clock about 1/5 of a second a day while with the atomic clock we are talking of seconds in millions of years.

Douglas Bateman brought the usage of Q in horlogerie to the forefront with an article that he wrote in the 1970s.  In that article he made a graph of all kinds of time measuring instruments, from cheap watches to the best atomic clocks.  These had many different kinds of oscillators and many different kinds of escapements.  On the graph he plotted the Q of the system against the rate accuracy the system provided.  The interesting thing is that for all the different systems the data points all fell pretty well on a straight line. This suggests strongly that the accuracy provided by a timekeeper depends strongly on the Q of the system and does not depend that much on the type of escapement or other factors.  The discussion about this finding continues to this day.   Mr. Bateman also published a series of articles on vibration theory as it applied to clocks in the Horological Journal of the BHI.

Mr. Douglas Bateman recently held a lecture at the Instant-Lab of the EPFL in Neuchâtel.  The well attended lecture presented many facets of the quality factor, its definition allowing its calculation starting from many different pieces of data.

Mr. Douglas Bateman at the Instant-Lab, Neuchâtel

After seeing this lecture I decided to measure the Q of the pendulum for the pendulum clock that I am presently building.  Conveniently, one of the definitions of Q is 4.53 times the number of complete oscillations of the system needed for the amplitude to go down by 50%.  This is easy to measure.

Below we see a graphic from data I recorded from my pendulum.  The scale at the bottom is the time in seconds.  At the left we can read the temperature, the relative humidity, the rate error per day.  At the right we can read the amplitude (line in blue).  As we can see I started the pendulum with an amplitude of about 1.6° and simply let it decay which produced a nice logarithmic curve.  The blips on the lines at times are measurement errors. Using this data I have calculated the Q.  At high amplitudes the Q is about 9'500, at low amplitudes the Q is about 11'000.  I attribute this difference mostly to the greater air resistance at higher amplitudes.




An interesting phenomenon can be seen at around amplitude 0.2°.  The amplitude drops quickly and at the same time the rate goes crazy.  I do not know what causes this, I presume some resonance of the pendulum support.  I will look into this further, but as it is at an amplitude well below the amplitude at which I expect the pendulum to work (the working amplitude will be above 1°) I am not worried about it.   This pendulum is just a suspension with a weight on a rod for the moment, it will be finished with temperature and barometeric pressure compensation in the next months.  I am hoping to increase the Q in the future with a better suspension as well.  The goal is the best atmospheric pressure pendulum possible.  Time will tell.