S4C10: The Watch

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a) The Mechanical watch

In S3 Section 9.a we exposed the historical relevance of knowing the time of the day as well as the time of the year. At this occasion we ran through the principles of the sundial, working off the shadows projected by a vertical stick called a gnomon to infer the angle of incidence of the light rays coming from the distant sun. Not very portable and not very accurate though. Furthermore, at the very same moment, this device for telling the time would suggest different times for two observers at different longitudes, thereby providing one more example that time is not absolute, including the way we partition it. Therefore, there needs to be a reference time shared by all users, and this is the principle behind coordinated universal time (UTC) to which the various time zones of the world are indexed. This is also the idea of chronometry, measuring the amount of time that has lapsed since a reference instant that happens to be defined as time zero or T0.

We have seen in the previous chapter, in S4 Section 9.c on the global positioning system, the tight relationship between space and time, something the theory of general relativity tells us as well by the way (refer to S1 Section 10.a), using the speed of light and the timing of radio wave signal transmissions and receipts to deduce differences in distance and time. Without relying on light speed, the ability to keep the time combined with the knowledge of the speed at which an observer is moving, say a man on a ship without any sighting of land, allows for the computation of the distance travelled and, if the latitude can be worked out from the stars, then the position on the possible circle centred at the origin (where T0 was taken) can also be calculated down to a point (two in theory but the navigator would know the cardinal directions towards which the ship is headed, say northwest as opposed to southwest, simply by observing the position of the sun and stars).

Hence, besides the practical aspects of being on time at a formal or personal meeting, the ability to keep time was essential in maritime navigation and avoiding train collisions. How can this be achieved?
The theory is simple, the realization less so. Besides a reference time, from a hardware perspective one needs an oscillator with a known frequency and a source of power to transfer energy to the oscillator in order to compensate the dissipation resulting from air and mechanical friction. In mechanical watches, the energy can be transferred manually by the winding of a watch which applies renewed pressure on a spring used to store this energy. This spring is called the mainspring and it is shaped as a spiral. When the user winds the watch he forces the spiral tighter and the spring releases force continuously by unwinding to its neutral shape. This winding process would need to take place every 2 or 3 days in the more sophisticated models and can even be replaced by self-winding mechanisms harvesting some of the kinetic energy the watch is subject to when we walk, for instance.

The watch, this is the name we give to the timepiece we carry on us, the English etymology may have to do with the watches on a ship, those work shifts during which a few persons would need to stay awake and alert to look out for possible trouble on the horizon, including changes in the weather, not just pirates. Back to the engineering of the watch, or the clock for that matter, which uses a suspended weight to store energy rather than a spiral torsion spring. We now have a source of energy so what we need is an oscillator, a way to slowly transfer energy to the oscillator, and the ability to read the time based on the number of oscillations.

The oscillator is called a balance wheel and it is undoubtedly the most valuable piece of the equipment as far as accuracy goes. It also uses the ability of a spiral spring (called hair spring or balance spring in this case) to drive the oscillation of a weighted wheel at a stable resonant frequency. The period of oscillation is a function of the wheel’s moment of inertia (the ratio of torque over the induced angular acceleration) and the stiffness of the hair spring. If you want to understand about this element and relationship better, I have included the link to the Wikipedia entry for balance wheel at the end of this chapter.

Next, we need a mechanism to transfer energy over several days from the mainspring to the balance spring so the oscillator can keep going, ideally forever. If we think it through, what we need is a way to have a tiny amount of force at the mainspring end translate into frequent tiny pushes at the oscillating end, the way we would regularly provide a push to a child on a swing to compensate for the various frictions involved. We are thus after a multiplier effect with discrete intervals, at least as far as the last mechanical interface with the balance spring is concerned. This is achieved through a series of indented cogs, where one revolution of the cog driven by the mainspring would drive hundreds of revolution of the last cogs in the series. This multiplier effect only requires having two circular set of teeth on the cog, one on the outside and one superimposed near the axis of rotation, and the multiplier coefficient is the ratio between both. Note that there is no magic, the multiplication in the rate of rotation doesn’t imply an increase in energy in the system once all the cogs are in motion and keeps their same rotation rate and thus their stored energy.

The mechanical complication is at the interface between the last cog and the oscillator, it takes the form of a two-pronged element called the escapement mechanism. Essentially, the upper part of the prong pushes down on the oscillator in one direction but the escapement is stopped by the bottom prong until the oscillator returns and pushes this bottom prong, and the upper one since they are tied together, back to their original position. Of course, there is more push transferred to the oscillator than is received from it, otherwise this would defeat the purpose, and each movement of the prongs creates the ticking noise of a watch or clock and, more importantly, allows the measurement of time by keeping the movement of the cogs to a specific angle at a known frequency.

From there, it is possible to have cogs moving at the desire pace to display seconds on one hand, and the movement of this cogs driving that of a cog used to measure and display the minutes on another hand, and the same goes for the hour hand on top of this. All that is left to do is encase all these mechanisms and layer a clockface with a dial to provide a quick way to read the time.

Perhaps this sounds ingenious but not hi-tech and yet, it is a marvel of engineering to have an oscillator remaining on pinpoint frequency when placed around our wrist and subjected to six degrees of freedom (an allusion to the gyroscope and accelerometer we discovered in S4 Section 9.b), i.e. any movement you can think of during the course of a day. Likewise, the ability to minimize friction to make the energy stored in the mainspring last for several days is truly impressive. And since you always wanted to know, the favoured material for highly accurate timepieces are jewels such as sapphire or ruby because they have a static coefficient of friction on steel that is 4 to 5 times lower than that of steel-on-steel.

b) Quartz chronometry

Jewel or not, high-precision mechanical watches are genuine art pieces, requiring both time and skill to manufacture, something one needs to pay for at the end of the day. This left the door open to technologies that could deliver at least as high an accuracy at a less expensive price point. Enters the quartz watch, first developed in the 1920s and then refined until it could become a consumer product in the 1960s, gradually relegating the mechanical watches to the realm of luxury goods.

Instead of a mechanical oscillator in the form of a balance wheel, the oscillation has to do with a phenomenon known as piezoelectricity which links mechanical deformation of material with the generation of an electric field, and vice versa. This reversibility means that by applying an external electric field to such a material with elastic property, in this case a silica crystal we call quartz, it will deform and then, in turn, generate an electric field as it returns to its original shape. Hence, it can be made to vibrate and the frequency of this vibration is a function of its shape, size and the crystal lattice orientation so this can be controlled in the quartz cutting process.

Typically, the resonance is engineered to correspond to a power of two, the favoured one being 2¹⁵ = 32,768 times per second. Then the principle is analogous to the wheels of the mechanical watch and each move creates a change of position or value in a binary switch, called flip-flops as per the registers in the CPU inner-memory (refer to S4 Section 2b). Every time the flip-flop returns to its 0 position a second flip flop will see its value shift from 0 to 1 and then, it returns to 0 the next time the first flip-flop value goes back to 0, after taking the 1 value in between. So every 4-oscillation cycle, the second cog will have gone through a 0-1 cycle which can feed a signal to a third cog that will go through the full cycle every 8 oscillations. This creates a digital counter that will increment the numbers displayed on your watch face by one second, or the cogs driving the seconds-hand by one tooth. By the way, one of the reasons why this 32KHz frequency is often used is because it lies slightly above the human hearing wave frequency band of 20Hz-20,000Hz.

Apologies but, as you should suspect by now, I can’t just leave it at that. We need to understand the piezoelectric effect a little bit better – not the details for the purpose of this section, just the high-level principles. All right, remember the composition of quartz as silica, well that is silicon dioxide, SiO₄ but the ratio is actually SiO₂ because each oxygen atom is shared. The creation of the electric field is the result of an asymmetry induced via mechanical strain such as compression where there was symmetry in electric charge in the default state. This would not occur if electrons were shared equally by the oxygen and silicon atoms. However, oxygen atoms are more polar, meaning they have greater electronegativity, and when compressed the “centre of gravity” of the negative charges in the silica will decouple from the centre of gravity of the positive charges, thus creating an electric dipole with a strength and orientation that is in part influenced by the extent of the mechanical stress applied to the material. When extended to the entire quartz volume, this dipole shift mathematically works out to a lining up of opposite charges at the crystal faces (hence the orientation of the quartz cut is important), which can drive electric current. That will do for our purpose. Oh, and the piezoelectric effect is what creates the spark your gas lighter relies on to ignite the gas.

c) Wearable computers

It would be easy to think of smartwatches as the next evolutionary step of watches, yet I think it would be a mistake because, let’s be objective, these devices are computers we happen to wear on our wrist and can also show the time and date, like watches. This is similar to the idea of the smartphones that we also happen to use for phone calls, occasionally.

The main difference between smartwatches and smartphones has to do with the elements within them but they both have sensors, a screen, a processing unit, and their own operating system. Because the combination of sensors and processing power is not identical, their usage is different. However, just like smartphones, these unique combinations of pre-exiting elements allow for the emergence of new use cases and applications. Some of these devices are meant to be used on their own, only connecting with a smartphone to upload some data for ease of visualization (dedicated technical sports watch for example) whereas others are much more dependent on the smartphone apps and can be seen as a hardware extension (that would be the more health and fitness-oriented watches) – of course, this is a continuum, not a binary separation.

You may want to refer to S4 Section 9.a on smartphones because, rather than repeating many aspects already covered, it is easier to state that the key differences at the moment are the absence of a camera system, biometric identification sensors and, generally, speaker, sound card and cell network transceivers were also omitted (though this is changing), resulting in a more independent device. Perhaps smartwatches are the future of smartphones provided we find an alternative display technology.

Now, let’s go back to health and sports activities. In S4 Section 9.b we looked at two types of sensors, the gyroscope and the accelerometer. These can be used to detect your strides or swimming movements, even discerning between different swimming styles (freestyle Vs breaststroke for example) and inferring the start of a new lap from a tumble turn. When coupled with a geolocation sensor (refer to S4 Section 9.c on GPS), you can record the distance you have hiked, biked or run.

When doing any of these last three activities one may also want to know the altitude one is at, as well as the cumulative ascent and descent from the start of the activity. This is the job of the barometer or altimeter to which the next section will be devoted. We probably also want to know our heart rate (HR) during the activity as a measure of aerobic training intensity and this HR data is also key in monitoring our health and stress level so we will look at it in section e).

d) Barometer & altimeter

The way change in altitude is measured is by detecting variations in atmospheric pressure; sounds simple enough but why is that? As in, why does pressure changes with altitude? There are two reasons.

The first – less important – is the change in gravitational interaction to which the molecules are subjected. This decreases as the distance from the centre of the Earth increases (gravity follows an inverse square laws so it is divided by four when the distance between two centres of mass doubles). Having a square relationship would suggest a dramatic shift in gravity except that the reference point is not the surface of the Earth but its centre of mass so with an average radius of 6,371km (it is 6,378km at the equator) an altitude of 2,000m above sea level is equal to 0.03% of the total distance, which translates into respective gravitational forces less than 0.01% apart.

The second reason is the air molecules are essentially stacked on top of each other with the entire column of air molecules located vertically above the pressure sensor weighing on it. This is the very same phenomenon we experience in a liquid such as the ocean where pressure increases with depth because there is more water above the diver.

Clear enough, so all we need to derive altitude is to know the reference pressure at sea level and the pressure measured by the sensor, right? Not so fast, this would be too easy, wouldn’t it? It so happens that sea level pressure is not a fixed quantity. Indeed, if you have read the third chapter of Series 3 on the atmosphere and the climate and in particular S3 Section 3.d on weather systems and phenomena, you would recall that anticyclones are high-pressure zones bringing good stable weather and, on the contrary, extratropical cyclones form around low-pressure systems and are harbingers of thunderstorms. Thus, the measurement of air pressure thanks to instruments called barometers, the technical name of the sensor I referred to earlier in this section, is key in weather forecasting and can be found in many homes and nowadays in some smartwatches.

What this means is that a drop in pressure can be the result either of an increase in altitude, or the sign that bad weather is coming, or perhaps both. How to tell? Well, the reality is you would most likely know whether your position is changing or likely to change along a vertical axis based on your activity. For instance, if you are running on a horizontal road, say a track in a stadium, there would be no change in altitude therefore your watch would know the change has to do with the weather and if you are hiking in a mountainous environment, it would most likely have to do with altitude. Depending on the watch model, you may be able to toggle between a barometric altimeter mode and algorithm and a standard barometer mode. Also, if your watch is equipped with GPS, this can further provide a datapoint about altitude and both, working together, enable the watch to quantify the respective parts played by the weather and altitude change in pressure variations.

There are a few micro-electromechanical systems (MEMS) that can do the job in a small package and some of them rely on the principle of measuring the displacement created by air pressure on a plate or membrane above a thin layer of gas. Does that mechanical stress concept sound familiar? It should because we have just seen it in section b) on quartz crystals and the miniature sensors rely on a phenomenon related to the piezoelectric effect called the piezoresistive effect whereby mechanical strain produces a change in the electric resistivity of a conducting material or a semiconductor and such a variation in an electric circuit can be measured. How small can such a sensor be though? Doing research for this section, I saw some advertised for smartwatches with dimensions of 2x2x1mm. Imagine that.

Furthermore, even though the inputs of such tiny and super-sensitive barometers are subject to noise in the form of wind gusts at the exact time the reading is taken, such inputs can be sense-checked by comparing them to other sensor inputs such as linear movements and rotations detected via an accelerometer and a gyroscope, respectively. No movement probably means the measurement of a sudden change in air pressure should be discarded.

e) Heart rate monitors

Knowing our altitude or cumulative ascent when training is a useful tool but really, in aerobic sports, it is nowhere as useful as knowing speed, power output or our heart rate. The latter indeed gives us a direct insight into the relative effort our body is experiencing because, if you recall S2 Section 1.d on the heart and blood vessels, the heart acts as a pump pushing blood containing oxygen throughout our body and these O₂ molecules are required for the production of ATP molecules (described in S1 Section 4.c), an indispensable enabler of muscle contraction. Thus, all else being equal, the higher the intensity of an effort, the more oxygen is required to sustain it (unless we tip in the anaerobic phase which some muscle fibres can handle though for less than 2 minutes in human beings because of the limited quantity of ATP that can be created without the help of oxygen). Accordingly, measuring one’s heart rate is a useful parameter in calibrating effort, whether it is in training or racing.

Unsurprisingly, our blood circulation is such a foundational facet of maintaining proper homeostasis that HR monitoring can be leveraged to track other aspects of our health. However, this section is about HR monitor, not the role of our heart so I will only mention resting heart rate and HR variability. The former can be an indicator of fitness, or lack thereof, though it is probably more useful as an indicator of relative stress and fatigue when comparing current measurements to a baseline measured over several weeks. As for HR variability, it is not purely a one-way indicator but, generally, greater variability is better, whether this relates to cognitive or physical functions.

Figure 6: Heart rate variability

Credit: YitzhakNat (CC BY-SA 4.0)

HR variability refers to the change in the duration of the intervals between heartbeats, from peak to peak. I Included an example of such intervals in Figure 6 above which is also useful in visualizing the pattern of human heartbeats and can be broken down into three phases. If you have read S2 Section 3.b on muscles contractions, you would recall this involves the depolarization of the membrane of muscle cells (they become less negatively charged). In an electrocardiogram this corresponds to the various waves and spikes as the atria and ventricles are being made to contract and then go through a repolarization phase before the next beat – I have included a link to the Wikipedia entry for electrocardiography in section g) if you want to read more about this topic.

This means the muscle cells responsible for the contraction, the cardiomyocytes, induce small electric fields on the surface of our body which can be detected by a pair of electrodes. This is the role of the chest strap, the most reliable hardware piece when it comes to measuring HR with output reading on a smartwatch. For a reliable detection of the electrical signals, the strap works best when it is in direct contact with the skin and after we start sweating because the wetting decreases the electric impedance and thus improves the signal level.

While a chest strap directly measures our heart rate, it is also possible to get an indirect reading by measuring our pulse. This is the alternative the optical sensors at the back of some smartwatches rely on. In this case we talk not about electrocardiography but about photoplethysmography (PPG). The complex name hides a relatively straightforward concept as far as biology and electronics go. When the heartbeats send pulses of blood through the blood vessels, this forces a small expansion of said blood vessels, all the way to the tiny arterioles and venules.

The way this is detected is by shining a LED into our skin and measuring not the absolute amount of light reflected but the variations. These changes are the result of the increase and decrease in blood volume as the pulse travels through the blood vessels and blood, like all other parts of our body, has a particular light absorbance and scattering coefficient. Even without knowing the coefficient precisely, it is possible to count heartbeats by using a photodetector to measure changes in the intensity of the light being reflected.

f) Trivia – the atomic clock

In the introductory paragraph of this chapter I reminded the reader that time is not absolute and in S4 Section 9.c on GPS there was a mention of atomic clocks being used as instruments of reference. The two aspects are linked and what we know as coordinated universal time is based on the International Atomic Time, abbreviated as TAI, which reflects the passage of proper time at the surface of Earth. This IAT is the result of the averaging of the time shown by several atomic clocks spread around the world.

So, how does an atomic clock work? No spring and wheels there to be sure and yet, frequency remains at the core of the concept of measuring the passage of time. The central idea is to tune the frequency of microwave radiations such that they correspond exactly to the inherent oscillation frequency of an atom, in which case it transitions from a low to high energy state, something that can be detected by measuring the electric current output of the clock circuit. The higher the current output, the closer to the resonance frequency.

For various technical reasons having to do with the principle of hyperfine transition, the selected chemical element for atomic clocks is Caesium and, as we saw in S1 Section 10.f on the base units of measurement, it is used to formally define the second as corresponding to the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9 192 631 770 Hz (i.e. per second).

The current precision level of existing atomic clocks is about 10^-15, meaning an error of 1 second every 10¹⁵ seconds, or about 32 million years and new technologies, including experimental optical clocks are pushing the accuracy even further, beyond 10^-17 and even 10^-18. This is obviously not a requirement in our daily life, even for GPS, but it does make a difference for some scientific observations and the ability to carry out new experiments.