The fork inside a watch

The frequency that gets an answer explains resonance through its quality factor, Q: the number of cycles over which a resonator’s stored energy falls by a factor related to $e^{2\pi}$, equivalently the ratio of its frequency to the width of its response. A car’s suspension has a Q of about two, a guitar string about a thousand. A resonator with a high Q rings for a long time, responds only to a narrow band of frequencies, and holds its frequency against disturbances, which makes it a clock. The pendulum clocks of the seventeenth century had a Q of a few thousand; the quartz watch on most wrists has a Q of tens of thousands, in a resonator the size of a grain of rice.

That resonator is a tuning fork, cut from a single crystal of quartz with its tines a few millimetres long, sealed in a tiny metal can, and driven by an electronic circuit through the piezoelectric effect: quartz develops a voltage when it is bent, and bends when a voltage is applied, so electrodes on the tines both drive the fork and sense it. The circuit keeps the fork ringing at its own resonant frequency, a chain of counters divides that frequency down to one pulse a second, and a tiny stepping motor moves the second hand. Since the 1970s quartz watches have been accurate to a few seconds a month at a cost of a few pence. Every part of that accuracy comes from decisions about the fork.

Fifteen halvings to a second #

The frequency was chosen backwards, from the circuit that counts it.

A binary counter stage, a flip-flop, produces one output pulse for every two input pulses, so a chain of them divides any frequency by a power of two. Two to the fifteenth is 32,768, and fifteen stages turn that frequency into exactly one pulse a second. Any power of two would do arithmetically; this one was chosen because it is the smallest power of two above twenty kilohertz, the upper limit of human hearing. A resonator at sixteen kilohertz would be audible, as some early electronic watches were, humming faintly against the wrist. One at sixty-five kilohertz would work just as well and cost more power, because every switching of every transistor in the counters dissipates a little energy, and a counter running at twice the frequency switches twice as often.

The watch’s whole power budget is a microwatt or two, enough for a small button cell to last for years, and most of it goes to the motor and the oscillator circuit. Choosing the lowest inaudible power of two keeps the divider’s share as small as possible.

The size the frequency sets #

The frequency also fixes the fork’s size. Each tine is a quartz cantilever, a beam clamped at one end, whose fundamental frequency rises in proportion to its thickness and falls as the square of its length.

A tine three-tenths of a millimetre thick rings at 32,768 hertz when it is 2.8 millimetres long: small enough for a watch, large enough to be cut and plated with electrodes by photolithography, the techniques of microchip manufacture. To ring at concert A, 440 hertz, a tine of the same thickness would have to be 24 millimetres long, and one of the frequencies below hearing would make the resonator larger than the watch. Going higher shrinks the fork, but by less than it costs in power, since frequency rises only as the inverse square of length. The quartz crystals in computers and radios, which need megahertz frequencies, are not forks at all but thin plates vibrating in shear, whose frequency is set by thickness alone; a watch’s fork exists because a watch needs a low frequency in a small space.

How sharply the fork answers #

The quality factor determines how firmly the fork, rather than the circuit, sets the frequency.

A resonator responds strongly only within a band of width $f/Q$ around its natural frequency, and within that band its phase swings through a half-cycle. The oscillator circuit keeps the fork ringing by feeding back a signal in phase with its motion, and if the circuit’s own phase shifts drift — with its supply voltage, its temperature, its age — the frequency at which the loop is in phase moves within the resonance’s width. With a Q of 80,000 the width is 0.41 hertz, twelve parts in a million, and a large phase error in the circuit moves the frequency by only a small fraction of that. With a Q of a thousand, the same error would move it eighty times further. This is the same effect the clock that is pulled into step describes for coupled oscillators: a resonator can be pulled away from its natural frequency by anything coupled to it, by an amount inversely proportional to its Q.

How long the fork keeps ringing #

What gives the fork its Q is its shape and its can.

A single vibrating tine, clamped at its base, pushes its clamp back and forth at every swing, and the clamp, attached to the rest of the watch, carries some of the vibration away. Two tines on a common base, vibrating in opposition, push their base in opposite directions at every instant; the forces cancel, the base stays nearly still, and almost no energy leaks into the mount. That is why tuning forks have two prongs, and why a musician’s fork, held by its stem, rings for many seconds while a single steel bar of the same size, held at one end, dies in a fraction of one. The quartz watch’s fork adds a second improvement: it is sealed in a can from which the air has been removed, because air damps a vibrating tine by viscous drag, and in vacuum the remaining losses are inside the quartz itself, which is among the least lossy materials known.

The figure’s representative values show the effect of each decision. A single clamped tine might ring for a few tens of milliseconds; a fork in air for about a tenth of a second; a fork sealed in vacuum for nearly a second, some twenty-five thousand cycles before its amplitude has fallen by a factor of e. The circuit has only to replace the tiny energy lost each cycle, which is why it needs only a microwatt.

Why a fork and not a bar #

The balance of the two tines is a case of a general principle about coupled resonators. The mass that makes another stand still shows how a second oscillator attached to a vibrating body can hold that body motionless at one frequency, by pushing back on it exactly as hard as the drive pushes forward. A fork does this to its own base: the two tines are a pair of identical oscillators coupled through the base, and their two normal modes are a symmetric one, in which both swing the same way and shake the base, and an antisymmetric one, in which they swing in opposition and the base is left still. The watch’s circuit drives only the antisymmetric mode, through electrodes arranged so that a voltage bends the two tines in opposite directions, and that mode cannot leak energy into the mount because it exerts no net force on it.

The fork was a watch’s resonator before quartz was. The Bulova Accutron of 1960 kept time with a steel tuning fork ringing at 360 hertz, driven by a transistor and two small coils, and it hummed audibly; it was accurate to about a minute a month, ten times better than a good mechanical watch, because even a steel fork has a Q in the thousands, far above that of a balance wheel. The first quartz wristwatch, Seiko’s Astron of 1969, used a quartz resonator at 8,192 hertz, and within a decade the industry had settled on the 32,768-hertz fork, made by photolithography in vast numbers. The shape survived the change of material, because the problem the shape solves — keeping the energy out of the mount — is the same for both.

Starting from nothing #

When a watch’s battery is first connected, the fork is not ringing, and the circuit has nothing to amplify except the fork’s own thermal motion: the random vibration that, as half a kT in a piece of wire shows for electrical circuits, every resonator carries at a temperature above absolute zero, with an energy of $kT$ in each mode. The circuit’s gain is set just above the fork’s losses, so the amplitude grows exponentially from that thermal level, by a factor e every few ring-down times, until the circuit’s limiting brings it to its working amplitude. Growing from thermal motion to a working amplitude a hundred million times larger takes about twenty e-foldings, and with a ring-down time near a second, a quartz watch takes some seconds to start after a new battery is fitted. The high Q that makes the fork a good clock also makes it slow to start, for the same reason it is slow to stop.

The parabola every quartz watch follows #

The remaining error is temperature, and quartz behaves in a way that the watch’s designers exploit rather than fight.

Every material’s elastic stiffness changes with temperature, and with it the resonant frequency of anything made from it; for most metals the frequency falls steadily as they warm, as the pendulum that must not grow when it warms shows for pendulum rods. Quartz is anisotropic, and its stiffness’s temperature dependence varies with direction in the crystal, changing sign in some directions. By cutting the fork’s blank at a particular angle to the crystal’s axes, the first-order temperature coefficient of the fork’s frequency can be made to vanish at a chosen temperature, the turnover point, leaving a parabola: the frequency is at its maximum at the turnover and falls on either side as the square of the departure, by about 0.034 parts per million per degree squared.

Two glasses that cancel a derivative describes the same strategy in optics, where two glasses are combined so that the first derivative of focal length with respect to wavelength vanishes, leaving a smaller second-order residual; it mentions in passing the AT cut of quartz plates, whose frequency–temperature curve is a cubic. The watch fork’s cut leaves a parabola, and the vertex is chosen near twenty-five degrees because that is close to the temperature of a watch worn on a wrist under a sleeve, which averages a few degrees warmer. On the wrist the watch is within a few parts in a million of its nominal rate, a tenth of a second a day. Left in a drawer at ten degrees it loses two-thirds of a second a day; worn on a winter walk at freezing it loses nearly two. Because the curve is a parabola, every departure makes it lose, so a watch that is worn sometimes and left off sometimes accumulates a steady loss that its owner corrects every few months.

Correcting what the parabola leaves #

Better watches remove the residual. A thermometer on the watch’s chip measures the temperature every few minutes, and the circuit adjusts the rate by an amount computed from the parabola — not by changing the fork, but by occasionally adding or dropping pulses in the divider chain, a technique called inhibition. Such temperature-compensated watches keep time to a few seconds a year. Radio-controlled watches go further and dispense with the problem: they set themselves each night from a time signal broadcast from an atomic clock, so the fork has only to be good enough to keep time between signals.

The ultimate limits of a quartz fork are aging and its own noise. A freshly sealed fork drifts by a few parts in a million in its first year as stresses in the crystal and its mounting relax and as minute contamination migrates on its surface, and then more slowly. The fork’s frequency also wanders randomly at very small amounts, a noise whose size is set by the Q and by the temperature, as the width that is a lifetime relates the width of any resonance to the time its excitation lasts. For a watch these are far below the temperature effect; for the oven-controlled quartz oscillators that discipline laboratory instruments and telecommunications networks, they are the dominant errors, and those oscillators hold their crystal in a heated enclosure at the turnover temperature to make the parabola irrelevant.

A resonator as a sensor #

The fork’s sensitivity to its environment makes it a sensor as well as a clock. A quartz tuning fork is the force sensor in many atomic force microscopes, where the shift of its resonance as its tip approaches a surface measures the forces between the tip and the atoms beneath; the high Q makes the frequency shift measurable at the level of piconewtons. The weight a frequency can find follows the same idea down to single molecules: anything that adds mass to a resonator, or stiffens or softens its spring, moves its frequency, and the sharper the resonance, the smaller the change that can be seen. Quartz-fork sensors measure gas density and viscosity, since the surrounding gas damps and loads the tines, and temperature, by deliberately choosing a cut with a large temperature coefficient instead of a small one. The watch fork is the one cut and sealed to be as indifferent as possible to everything except its own elasticity.

A fork drawn as two beams #

The tine formula is that of a uniform clamped cantilever of isotropic material with representative values for quartz’s Young’s modulus and density, whereas real quartz is anisotropic and the fork’s shape is more complicated than a pair of rectangular beams; it gives the right size to within the ranges quoted. The quality factors are representative values for the three cases, not measurements of particular devices; real watch forks have Q between about twenty and a hundred thousand. The resonance curves are those of a single damped driven oscillator. The temperature parabola uses the typical coefficient of −0.034 ppm per degree squared and a turnover of 25 °C, which vary by a few per cent and a few degrees from fork to fork; the wrist temperature of 31 °C is illustrative. Aging, supply-voltage effects and the stepping motor’s own timing are not drawn.

Still open: whether quartz will be replaced in watches #

Micro-electromechanical resonators made of silicon, etched on the same chips as the electronics, now compete with quartz in many timing applications. Silicon is not piezoelectric and its stiffness changes strongly with temperature, so silicon resonators need active compensation and electrostatic drive, but they can be made smaller and integrated directly. Whether they can match the quartz fork’s combination of microwatt power, high Q and long-term stability at the price of a watch — or whether the quartz fork, invented in its watch form in the 1960s and essentially unchanged since, will remain the cheapest way to count seconds on a wrist — depends on manufacturing more than on physics.

The habit worth carrying away is to ask what each number in a design is doing. A watch’s fork rings at 2¹⁵ Hz because fifteen halvings make a second and that is the lowest power of two the ear cannot hear; it is 2.8 mm long because a quartz tine’s frequency falls as the square of its length; it is a fork, sealed in vacuum, because opposed tines and no air lift its Q from thousands to tens of thousands; and it loses only a tenth of a second a day on a wrist because its parabola of 0.034 ppm per degree squared is centred at 25 °C. A quartz watch is a resonance designed around a person.