Clocks, piezoelectric crystals and logic gate solutions - part 1

Lying in bed at night, I find myself considering exactly how the cheap analog clock in front of me keeps such perceivably accurate time, day after day and night after night. The only thing which can halt the unstoppable passage of the ticking hands is the slow discharging of the battery - but how does the input of voltage lead to a constant and reliable motion, which doesn't degrade as the voltage of the battery slowly decays?

Piezoelectric crystals
The piezoelectric effect describes how the application of mechanical stress to certain crystalline materials can generate a small electric current - the pressure causes the positive and negative charge centres to move, with the result that a weak external electric field is created. Early experiments showed that the effect was exhibited in a variety of natural crystals including cane sugar, topaz and quartz, with more recent developments augmenting this range with man-made structures including barium titanate (BaTiO₃₎ and lead zirconate titanate (Pb[ZrₓTi₁₋ₓ]O₃). [1]

As an aside, the latter of these two compounds is classified as an "intermetallic compound", a curious label which means that metallic bonding occurs with defined stoichiometric ratios (i.e. for x lead atoms there are y zirconium and z titanium atoms, and for 2x lead atoms there are 2y zirconium and 2z titanium atoms) - within mainstream education I have never heard of such materials which form quasi-ionic lattices, so I find this very interesting. In the case of lead zirconate titanate, the chemical formula implies that for every lead atom there are x zirconium atoms, (1-x) titanium atoms and 3 oxygen atoms (where x is between 0 and 1 inclusive). Having thought through the implications of this formula, it would appear that the crystal is not as uniform as giant ionic structures are. The most common form is PbZr_0.52Ti_0.48O₃, telling us that the ratio of the number of PbZrO₃ unit cells to PbTiO₃ ones is 0.52 : 0.48, or 13 : 12 [2] (since for the two statements Zr = x and Ti = 1 - x to be valid over the restricted range [x, (1 - x) ∈ ℕ], the two solutions are [Zr = 1, Ti = o] and [Zr = 0, Ti = 1]).

This isn't the first time my attention has been drawn to the idea of these materials - while I was sailing with my family, lighting the gas stove made me consider exactly what causes the spark. It turns out that piezoelectric crystals are indeed involved: the movement of the trigger causes a spring-loaded hammer to strike the crystal, inducing a voltage by the piezoelectric effect which produces a spark against a metal plate. This high pd is enough to ignite the fuel, lighting the burner. 

The converse piezoelectric effect works by using a voltage to cause variation in the width of a piezo crystal. Since large voltages induce only tiny changes in the width of the crystal, the effect can be exploited to make motors which move objects with large precision - the crystal element is held against the target and a potential difference is applied across is, causing it to minutely change in width and push the target along by microns at a time. These motors are patented and manufactured by NanoMotion.

So how does this relate back to clocks keeping time? Well, these magical materials (usually quartz in this case) have a voltage applied between opposite faces by the battery, in turn causing the width of the crystal to oscillate thousands of times a second. However the crystal will not simply resonate by connecting the battery, but instead the electrical output needs to be fed back into it to continue the oscillation [6]. Since quartz oscillates at a fixed frequency of exactly 32768/s [3], a control circuit can convert these oscillations into even ticks of the clock by using a frequency divider circuit: these are used to take a high frequency signal in, and realise it to a much lower frequency signal - you want the clock to tick once per second, not 32768 times per second! Frequency divider circuits come in many different forms, but the simplest involves a chain of T flip-flops. These start with a default voltage of logic 1 or 0, and with each defined input will switch between the two. For example if the default was logic 1, and it will only flick to the alternative logic value when it receives an input of logic 1, the D flip-flop will output a digital signal of half the frequency. Therefore, the frequency-reduction factor of the circuit is 2^{number of flip flops} [5]. Conveniently, 32768 = 2¹⁵, so a circuit of fifteen sequential D flip-flops will reduce the frequency down to 1Hz.

Next, this signal will be fed into a stepping motor, which will use each electrical 'tick' to cause a mechanical tick of the clock: this type of motor is very useful for precision control of rotations. It is composed of a rotor with n metal teeth, and a stator with (n - 2) teeth connected to 8 solenoids arranged in a circle. The image below shows this well:

This is a view of the system from only one end. If we took the rotor out and looked at its cylindrical length, it would look like this:

In each step, two opposite solenoids will have positive magnetic charge and the two solenoids perpendicular to them will have negative charge. This will cause the rotor to jump by a quarter of a gear each time the field rotates [4]. It is better explained with my animations below; the first shows the process slowly, and the latter shows the net effect by playing it through at a faster rate. Since the rotor has 30 teeth and each 45° field rotation causes a quarter-gear jump, a full turn of the motor will occur every 120 field rotations - therefore my clock would run twice as slow as a normal clock, so real ones would have only half the tooth values (and be fed with a logic 1 frequency of 1Hz).

Sources

[1] NanoMotion - "The Piezoelectric Effect" - here
[2] Wikipedia - "Intermetallic" - here
[3] Explain That Stuff - "Piezoelectricity" - here 
[4] Youtube - digitalPimple - "Stepper Motor Basics and Control - How it works" - here
[5] Stack Exchange - Forum - here
[6]Hackman's Realm - "Information on Electronic Quartz Crystals" - here