And Now, A Word On Quality

The Quality Factor supposedly is a one-stop shop for quantifying precision. But is it the whole story?

More than one kind of Quality: Roger Smith Series 1 movement, with co-axial escapement

If you’re really into precision and accuracy, chances are, you’ve heard of something called the Q factor – Q for Quality. The Q factor of an oscillator – a balance, a pendulum – is supposed to be a simple formula which captures the essence of what makes a watch accurate. But it turns out the Q factor isn’t so simple and that, while an essential tool for understanding why a precise watch is actually precise, it’s not the whole story. It is, however, a very big part of the whole story and to understand what it is, is to understand something truly fundamental about watchmaking, and why watches have evolved in the way that they have.

In Pursuit Of Precision

Precision timekeeping is a funny thing. Watchmaking for most of its history was something of an empirical art – watchmakers had useful rules of thumb which got passed down from one generation to the next, and there were certain basic principles which began, over the centuries, to emerge and which were adhered to by anyone serious about precision.

The two most basic principles of all the basic principles are these:

• The oscillator should have a natural frequency, and

• Interfere with the oscillator as little as possible.

Every clock and every watch has an oscillator with a natural frequency. Exceptions to this rule are devices like water clocks, which measure the passage of time based on the continuous flow of water out of a container, or candle clocks, which show time as the rate at which a candle burns down (the time-honored, hahaha, hourglass is also a continuous process timekeeper as is, for that matter, the Congreve clock, which measures time with rolling balls, and for which the jokes write themselves). In a pendulum clock, the oscillator is the pendulum and the beauty of a pendulum is that the natural frequency is proportional to the pendulum’s length. In standard gravity at sea level, a pendulum 0.944 meters in length will complete one swing per second and moreover within certain limits, the frequency is independent of the size, or amplitude, of the swing (in practice this is only an approximation which works only for low amplitudes, but it’s close enough for government work, as a friend of mine who works for the IRS liked to say).

An oscillator which takes the same amount of time to beat regardless of the size of the swing is said to be isochronous, and it’s the property of isochronism that makes for a precise timekeeper. The pendulum is isochronous because the restoring force (gravity) is proportional to the size of the swing; in a watch, the same is true of the balance except the restoring force is the balance spring, not gravity. (This is simply to say that gravity pulls back harder as the pendulum oscillations get larger – a phenomenon familiar to any stay-at-home Dad who has gotten whacked in the face by a playground swing. Ask the man who knows.)

Very early on, clockmakers recognized that pendulum swings needed to have small amplitudes for the best accuracy and empirically, they also discovered that a massive pendulum swinging at a small amplitude was the ideal design. In watches, the balance beats at a higher rate – 18,000 vph, traditionally, although modern watches usually beat at 28,800 vph or faster. A large balance with a heavy rim and lightweight arms was the standard for precision watches for many years but it was discovered that you could get results as good or better with a smaller balance driven at a higher rate.

These seem like contradictory approaches but they are based of course on the fundamental difference between a clock and a watch, which is that a watch is in motion and a clock is not. Unlike a pendulum, a balance is isochronous at higher amplitudes and loses that property as amplitude drops.

What ties both approaches together, is a concept that takes mechanical horology out of the realm of empirical knowledge and into the the realm of quantifiable science. This concept is the so-called Q factor.

Q-Factor And Precision

When I first read about Q factor I thought I had found the key to understanding exactly why some clocks and watches are better than others (by “better,” I obviously mean more precise, not more visually entertaining or prestigious). The concept comes originally from electrical engineering and was first used in 1914. The basic definition of Q is that it is the ratio of stored energy in an oscillator to the amount of energy lost per oscillation. If you lose a lot of energy then your oscillator is heavily damped; if for some reason it’s damped enough that all energy is lost in one swing then the ratio of energy stored to energy lost is 1/1. A pendulum swinging in oil (or trying to swing, anyway) is a possible example. At the other extreme is a pendulum swinging in a vacuum on frictionless bearings – once set going, it will swing forever with a Q of ∞. That pendulum is under-damped. NIST (the National Institute Of Standards And Technology) offers a slightly different definition:

“The quality factor, Q, of an oscillator is defined as its resonance frequency divided by its resonance width. Obviously a high resonance frequency and a narrow resonance width are both advantages when seeking a high Q. Generally speaking, the higher the Q, the more stable the oscillator, since a high Q means that an oscillator will stay close to its natural resonance frequency.”

The association of Q factor to mechanical watchmaking is something we owe to Douglas Bateman, who popularized Q as a way of objectively quantifying the precision of an oscillator. TheWatches.TV interviewed Bateman on the subject.

The association of high Q with higher precision seems obvious from calculating the Q of different timekeepers:

• Mechanical watch, 300

• Tuning fork watch (Accutron), 3000

• Quartz watch, 10,000

• Shortt-Synchronome high precision pendulum clock, 110,000

• Oven controlled quartz oscillator (kept at constant temp), 1,000,000

• Optical mercury ion atomic clock, 1,000,000,000,000,000

Not coincidentally, oscillators with higher Q factors mostly have higher Q frequencies, however the Shortt-Synchronome clock used a seconds pendulum, so higher frequency is often but not always synonymous with higher Q. Here it’s useful to remind ourselves that Q in mechanical systems is the ratio of total energy to energy lost per oscillation and you can get there by increasing frequency or by decreasing friction and increasing mass – or some combination of both. You can get higher total energy by driving the oscillator at a higher frequency but you can also do it by having a massive oscillator beating slowly – the bob on a Shortt clock was a cylinder of Invar weighing 14 pounds.

Shortt-Synchronome clock, 1929

Now for all the time I’ve known about Q, it never occurred to me until recently that Q is not the whole picture. For sure an oscillator with a high Q factor is inherently more stable than one with a low Q factor – that is, it will over time drift very little from its natural resonant frequency. But it’s not the whole picture. An oscillator has to be driven by something and that something is the escapement. Escapements are designed to provide consistent driving force but no escapement is perfect and there are inherent slight variations in even the best escapements. No oscillator system is frictionless and frictional losses are inherently variable from one oscillation to the next; air turbulence will be another variable (one reason why high precision pendulum clocks were often in vacuum enclosures).

As it turns out, these sources of “noise” in the system can be quantified as well. If we’re talking about a pendulum clock, the total energy in the system varies (for the above reasons) but you can calculate the expected mean (average) energy; call it E. You can then get another ratio: The ratio of mean energy to the overall expected deviations. There is a fantastically concise description from the Leap Seconds blog, which as far as I can tell is where the description of this second ratio first appeared:

“To summarize, E is the total energy of the pendulum. Measure it very closely and you can calculate the mean, ∆E, and the standard deviation, σE, of energy gain/loss. Note σE is the combination of all possible sources: friction in support, suspension, and air drag, as well as impulse timing, duration, and power. So from these three energy values, it's easy to compute two ratios. E/∆E is what we call Q and ∆E/σE is what we call P. That's it.”

If Q stands for Quality then P can stand for Purity – that is, how noise-free or noisy the oscillator is. The argument in the article is that in a physical clock (not an idealized oscillator, which is a subtle point but key to the argument of the whole article) you need both, but neither is necessarily a one-size-fits-all solution to precision. A pendulum with a high Q might not perform very well if a lot of variation exists in the action of the escapement, changes in barometric pressure, and so on and one with relatively low Q might still perform reasonably well if P is high. Douglas Bateman says as much in the video interview – an oscillator can have an inherently high Q factor in design but you can screw it up in execution of you’re not careful.

One well-known source of noise in the system is oil, especially oil on the impulse surfaces of the escape wheel and lever. As lubricants age, the amount of energy delivered from the mainspring to the balance changes. Ideally, an escapement should have no oil on the impulse surfaces – the impetus behind the invention of the co-axial escapement.

Watch enthusiasts often say that all other things being equal, higher frequency will give better precision and it seems as if this is partly due to higher Q factor, but also due to higher frequency offering better resistance to disturbances. As you increase frequency, the frequency of movements (arm swinging) that might disturb precision are further and further away from the natural resonant frequency of the balance. A pendulum clock avoids this by being stationary but they make lousy wristwatches. It’s partly for this reason that when John Harrison finally made a successful marine chronometer, it had a balance, not a pendulum – and a whole host of features designed to reduce variations in the power of the escapement (a Remontoire, diamond pallets) variations due to temperature (temperature compensation) and so on. As with any real-world system, with watches, it’s not just what you do, it’s how you do it … as it turns out, you must mind your Ps and Qs.

Thanks to Leapsecond.com, an old-school hidden horological treasure trove. For NIST on Q factor, check out their glossary, Time And Frequency From A-Z.