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Quartz Crystals Theory
[Frequency Control Products / Quartz Technology]
Technical introduction
In their investigations the brothers Pierre & Jacques Curie found that electrical charges were produced by mechanical stresses applied to various naturally occurring crystals. This phenomenon is called the piezo-electric effect, being derived from the greek piezein = to press. Conversely it was found that the crystal was deformed by the application of an electrical charge, later the application of an alternating electrical field introduced mechanical vibrations. The quartz crystal behaves as an electrical resonance circuit with very low attenuation
Quartz was found to be one of the naturally occurring crystalline substance that exhibited the piezo-electric effect and being very stable, both chemically and mechanically, it became of great interest to the early electronic experimenters. The chemical description of Quartz is Silicon Dioxide SiO2. Although approximately 14% of earth's crust consists of SiO2, it occurs relatively infrequently in usable crystalline form with the necessary purity and without physical defects, cracks etc. Raw quartz of quality suitable for the production of quartz crystal units is found in Brazil and Madagascar. For this reason continuous attempts have been made to produce quartz synthetically; that is to grow by recrystallization on to quartz seed plates. These processes are now commercially successful. Quartz chippings are dissolved in an alkaline solution in steel autoclaves at approximately 400°C and a pressure of 10000N/cm2, the desired direction of growth is ensured by the careful preorientation of the quartz seed plates. The growth rate is normally approximately 1mm/day. Slow growth-rates result in a more homogeneous material since fewer foreign atoms are being incorporated into the crystal lattice.
Synthetic quartz is now grown in suitable length and cross sections to enable the subsequent processing steps to be carried out in the most efficient way, with minimum cutting losses. Grades available today are very high purity and mechanical quality, so that synthetic quartz is used almost exclusively for the production of quartz crystal units.
AT Cut
Its outstanding physical properties have made the AT-cut the most used cut for oscillator crystal production. The position of the AT-Cut relative to the crystal axis can be seen from the figure below
Oscillator crystals in the AT-cut are usually manufactured in the range 800KHz to 360MHz. AT-cut crystals are thickness shear vibrators. The crystal vibrator is usually a round disc. The thickness (d) of the disc is related to the fundamental mode frequency by the equation :
F(kHz) = N / d(mm)
where N=1660kHz/mm , for AT cut
The crystal vibrator has thin metal electrodes deposited on both sides by evaporation, these electrodes are the means whereby the alternating electric field is applied thus stimulating the mechanical oscillation. With overtone oscillation only odd-numbered harmonics can be generated. With an even-numbered overtone the electrodes would show identical polarity and consequently no electric field would be developed for the stimulation of mechanical oscillation.
Normal overtone oscillations are 3rd, 5th, 7th and 9th, the upper harmonic oscillation is not exactly an integral multiple of the crystal fundamental frequency but can differ from this up to some percentage.
The principal advantage of AT-cut over other cuts is the low frequency sensitivity to change in temperature. It follows a 3rd degree curve with inflection point which lies between 25°C and 35°C depending on the actual cut angle and mechanical construction.
Df/f = A1 * (T-Tref) + A2 * (T-Tref)2 + A3 * (T-Tref)3
Such equation can be reduced, if one refers to the inflection point temperature Tinv instead of the initial temperature Tref
Df/f = a1 * (T-Tinv) + a3 * (T-Tinv)3
Where :
a3 = -1.05 E-4
a1 = 0.085 * Df
Df = fzz-f0 (expressed in one-sixieth of one angular degree)
By choice of the appropriate cutting angle, two inversion points appear in the curve, a maximum below 25°C and a minimum above this temperature. At each of these inversion points the temperature gradient is zero. This property is, at the upper inversion point, utilized where crystals are operating in a thermostatically controlled environment (oven), judicious choice of the cutting angle will cause the zero temperature gradient point to coincide with thermostat temperature, thus the thermostat or oven temperature must be clearly stated when ordering crystals for particular use.
From the figure it can be seen that the symmetrical ranges of temperature with respect to the inflection point are meaningful. A one sided limitation means no relief for the manufacturer in general, due to the symmetry of the curves the suggested tolerance is only observed with opposite signs. Cutting angles of approximately 10 arc seconds can be accurately maintained, however other mechanical effects can influence the variation of frequency over the temperature range, thus technical as well as physical limitations are imposed on minimizing the temperature coefficient. For the determination of the frequency temperature coefficient it is sufficient to measure the frequencies at three temperatures. For higher accuracy five measuring points or more may be necessary. By means of this, the best adapted cubical parabola is applied and the appropriate coefficients a1 and a3 determined
A correctly designed resonator should produce a smooth curve for its temperature coefficient with the value of a3 within +/-20% of the value indicated above. In the case of other temperature ranges the accuracy of the curve can be increased by adding a 4th order term in the equation
Equivalent circuit and motional parameters
The electrical properties of the oscillator crystal as a function of the frequency can be depicted in the vicinity of the resonance frequency by an equivalent circuit diagram.
The oscillating mass is symbolized by the dynamic inductance L1, whereas the elasticity is represented by the dynamic capacitance C1. Overtone resonances, as well as spurious resonances, can be depicted by additional parallel connected resonance circuits. At higher frequencies (above 100Hz) a parallel conductance value G0 may be required for modelling the behavior. The insulation resistance is also modelled in this. The insulation resistance may be impaired by the following influences :
– fine leak of the glass feedthrough due to insufficient glass quality
– long-term effect : through the migration of electrode material along the so-called etching channels which can form due to non-homogenities in the crystal growth. This migration can only occur when a DC voltage is applied between the crystal pins over an extended period (several years) and has so far only been observed occasionally. The insulation resistance can collapse down to a few kW. The result of this is that the DC operating point of the capacitance diode connected to the crystal is displaced in VCXO circuits, which in turn leads to a greater frequency shift occurring. It is therefore recommended to operate quartz crystals free of DC voltage by connecting a blocking capacitor in series to the crystal unit
In addition a parallel capacitance Cp, which differs from static capacitance C0 measurable at low frequencies, acts at high frequencies owing to the influence of the lead and leakage reactances.
The value of the dynamic capacitance C1 is very small in comparison with capacitances normally found in oscillating circuits, as it it is in the range of a few fF.
C1(f) = 0.1 * Kc del2 (mm2) *fs(MHz)/n3
fs = resonance frequency
del = electrode diameter
n = ordinal number of the harmonics
Kc = correction constant
1 for fundamental oscillation
0.85 for 3rd overtone
0.75 for 5th and higher overtones
With a given resonance frequency it can be seen that the C1 can be varied by changing the electrode diameter, this variation is limited by the actual diameter of quartz vibrator. L1 and C1 are related by Thomson's formula :
L1 = 1/(w2*C1)
The dynamic resistance of the crystal unit R1 represents the mechanical losses due to molecular friction within the resonator plus the damping induced by the mounting system and the acoustical damping of the gasfilled housing. The static capacitance C0 represents the capacitance between the evaporated electrodes using the quartz resonator material as the dielectric
C0(pF) = 0.02 * del2(mm2)*fs(MHz)/n
Thus the capacitances of the mounting system and the casing have to be added (0.5pF to 2pF) , which leads to a new equivalent circuit.
Depending on the particular enclosure type C0 normally lies between 1pF and 9pF, oscillator crystals are normally designed with C0 less than 7pF. Its value can be influenced by an appropriate electrode diameter. For exact measurement of C0 it is essential to indicate if the measurement must be made with the crystal housing grounded or not. Likewise the frequency at which the measurement of C0 is performed must be off the resonance frequency of the crystal to be measured. Measurement near to resonance frequency indicates the parallel capacitance Cp which may differ from C0 particularly at high frequencies.
The quality factor of quartz crystal is specified as below :
Q = 2pfsL1/R1 = 1/(2pfsR1C1)
The curve below shows the reactance of a lossless crystal resonator. There are 2 resonance frequencies, the series resonance frequency fs and the resonance frequency fp.
fs = 1/(2p*(L1C1)0.5)
fp=1/(2p*(L1*(C1*C0)/(C1+C0)))
The series and parallel resonance frequencies are related by the formula :
(fp-fs)/fs = 0.5 * C1/C0
However, for a lossy crystal the dynamic branch of the equivalent circuit comprise L1, C1 and R1 which are in parallel with C0. Thus the impedance of this combination is only real at the resonance frequency fr above fs.
The complex impedance is given by :
Zs=R1*jX0/(R1+jX0) where X0=-1/(wC0)
The series resonant frequency of L1 and C1 is not the only frequency of the quartz crystal, as can be seen from the polar diagram representing the admittance of the circuit
The most important frequency for practical application is the resonance frequency fr at phase 0. The measurement technique in accordance with IEC444 also refers to this frequency. The frequencies fa, fp and fn lying at the high impedance end of the circle cannot be excited in oscillators. The centre of the circle is defined by the angle :
fm = arctan(2R1wC0)
and its distance form the real axis given by :
B0 = wC0
With increasing frequency the angle between the centre of the circle and G-axis increases, thus the frequency differences between fr, fs and fm also increase. Likewise the resistance Rr at zero phase also increases in comparison with R1. The so called figure of merit
M=1/(wC0R1)
Shows on the diagram as the difference between fr and fs (or Rr and R1).
At frequencies above 120MHz, M can become smaller than 2, so that locus no longer cuts the G-axis, thus fr disappears and is no longer seal.
Quartz crystal with load capacitance Cl
Fundamental mode quartz crystals are normally operated with a load capacitance, which allows the circuit capacitance variations to be compensated. The load capacitance is normally in series with the resonator and this causes it to oscillate at a new frequency, which is sometimes called "parallel resonance frequency", even though the load resonance frequency fl should always lie far nearer the resonance frequency fr (or fs). Than parallel resonance frequency fp. Overtone crystals in application without frequency modulation and FSK are often used at series resonance without load capacitance as the circuit normally contains a tuned (LC) circuit and the necessary compensation can be achieved by slightly detuning this element.
Figure below shows the reactance curves for a quartz crystal, then with a laod capacitance in series
fl=fs(1+C1/(2(C0+Cl)))
Rl=R1(1+C0/Cl)2
If one defines the difference in frequency fl and fs as load resonance frequency offset Dl, then we have
Dl=(fl-fs)/f = C1/(2*(C0+Cl))
The pulling range of the element is defined as the change in frequency produced by changing the load capacity from one value to an other :
Finally We can define pulling sensitivity S as the frequency change in ppm per pF change in the load capacitance
S=C1/(2*(C0+Cl)2)
From the above formulae we can deduce :
With low load capacitances, the change/pF would make S very large and thus the practical minimum value lies in the range between 8 and 10pF. Conversely with high load capacitance the pulling sensitivity is very small, so probably 100pF represents approximately the upper limit
Spurious resonances
All crystals have got spurious resonances (unwanted resonance responses) besides the main resonance frequency. The ratio of spurious resonance resistance RNW to resonance Resistance Rr of the main wave is generally specified in the attenuation constant dB and is designated as spurious attenuation :
A = 20*lg RNW/Rr
Values in the range 3 to 6dB are normally sufficient. For filter cyrstals attenuations greater than 40dB are often required. This can only be achieved by special design techniques and involves the use of very small values of the dynamic capacity C1. The achievable attenuation decreases with higher frequency and with higher orders of overtone. It is found generally that plano-parallel quartz resonators have better spurious attenuation than plano-convex or bi-convex resonators. In specifying spurious resonance parameters it is necessary to give anindicaiton of both the acceptable attenuation level desired and their frequencies relative to the main resonance frequency. In AT-cut mode the so-called un-harmonic resonances exist only above the main resonance in the region of +40KHz to +150KHz for plano parallel resonators, between +200KHz to +400KHz for biconvex or planoconvex resonators.
Drive level (DLD)
The amplitude of mechanical vibration of the quartz resonator increases proportionally to the amplitude of the applied current. High drive levels lead to the destruction of the resonator or the vaporization of the evaporated electrodes. The upper limit for drive level is approximately 10mW. As the reactive power oscillating between L1 and C1 is represented by Qc=Q*Pc, for Pc=1mW and with a Q of 100000, Qc is equal to 100Watts. The oscillation amplitude can be exceeted with relatively low level of drive Pc, thus resulting in the crystal frequency moving upwards. This frequency dependence on drive level is more pronounced with increasing overtone order. A well performing crystal should start to oscillate easily and its frequency should be virtually independent of the variation of drive level from a starting level of about 1nW. In todays semiconductor circuits with very low power consumption the crystal has to work well laso at very low drive levels. Crystals that have badly adhering electrodes or on which the surface of the resonator is not fine enough, will show higher resistance at low drive levels. This effect is called Drive Level Dependence. Usually production tests of DLD are performed between 1 to 10 microwatts and then at 1mW. The relative change in resistance is then used as the test criterion
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