Quartz crystals
(This is an adapted version of part of an article I wrote for the Dutch amateur radio magazine Electron, August 2016.)
Quartz crystals have been used in radio technology since the 1920s, to build very stable oscillators and to filter signals. In WW2 they started to be mass-produced, as is very nicely documented in [1], back then based on natural quartz from Brazil. Nowadays quartz crystals are still a mass product (more than 2 billion per year, according to Wikipedia), most of which spend their lives humbly clocking a microprocessor. But the importance of even that task should not be underestimated: according to [2] a train crashed in 1972 due to a badly designed crystal oscillator spontaneously jumping to its third overtone.
Such a crystal is nothing more than a slice of quartz, sawn under the right angle from a (nowadays synthetic) crystal, with an electrode on both sides. The picture shows two common crystals with frequencies of a few MHz; usually such crystals are in metal enclosures, but sometimes one encounters these glass versions. These crystals use the so-called thickness-shear vibration, in which both surfaces of the crystal plate shift w.r.t. each other, as sketched at the right. The two crystals above are much larger and resonate at lower frequencies, alternately getting longer and shorter. There are also crystals which resonate by bending like a tuning fork; among others the small 32.768 kHz crystals in watches work like this.
The next figure shows the well-known equivalent circuit of a crystal, with in blue typical values for a 10 MHz crystal. The inductance L m and the capacitance C m are called the "motional inductance" and "motional capacitance", because they are directly related to the motion of the crystal (as will be discussed further down the page). C p is the parallel capacitance: the electrodes on the crystal form a small capacitor. And R m is a resistor which represents the losses (due to friction etc.). The frequency on which the series circuit of L m and C m resonates is called the series resonance: at this frequency the crystal has a very low impedance, practically only R m (with C p in parallel, but that has little effect). In contrast, the impedance becomes very high at the parallel resonance frequency: there L m resonates with the series circuit of C m and C p connected in parallel to it. Their total capacitance is slightly less than C m alone, so the parallel resonance frequency is a little higher than the series resonance. In practice, parallel to C p there is the capacitance of the circuit to which the crystal is connected, which does not have an influence on the series resonance, but does affect the parallel resonance. That is why crystals for parallel resonance are always specified for a specific extra parallel capacitance; in this example, 25 pF is needed to bring the crystal to 10.000 MHz, as shown in dotted lines.
How does a quartz crystal resonate?
At first, the answer to this question is simple: quartz is slightly elastic, so it can vibrate in the same way e.g. a rubber band or guitar string can vibrate. You start with pulling the rubber band from its neutral position. Due to this deformation, a force arises in the elastic material which tries to undo the deformation. Next, you let go of the rubber band. The material starts to move towards its neutral position, the force decreases and becomes zero when the neutral position is reached. By then however, the material is in motion at some speed and will "overshoot" to the other side of the neutral position. The elastic force will work against this, and so on. We typically don't imagine a crystal as elastic, but the principle is the same. Only the force becomes very large already at a very small deformation, so the vibration will be much faster (higher frequency) than with a rubber band.
However, in a quartz crystal there are not just mechanical effects as discussed above; due to the piezo-electric effect, also electric effects play a role. On both sides of the crystal there's a metal layer, between which a voltage can exist, and/or on which a charge can accumulate. The relationship is illustrated in the figure. We see the crystal as a "black box", with two kinds of "input" at the left and two kinds of "output" at the right. The inputs are how much the crystal is deformed, and how much electrical charge has been put onto the connection plates. The outputs are the force in the crystal which acts against the deformation, and the electrical voltage we measure across the crystal. How these outputs depend on the inputs is determined by physical properties of the crystal, and is indicated in the figure.
B.t.w., this model is not the only possibility. One can also choose different combinations of input and output, e.g., take the force as input and the resulting deformation as the output, and there is also some freedom in the choice of plus and minus signs (i.e., which direction is denoted positive). But the model as shown here, is the handiest for explaining the resonance.
Let's first assume that there is no charge on the crystal. The crystal is not connected to anything, but we do deform it. We see that that deformation causes two effects: a mechanical force and and an electric voltage. That mechanical force gets a minus sign, indicating that it works against the deformation.
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