Quartz crystals are generally suited for the manufacture of frequency selection or frequency control devices. In oscillators with load capacitance in series with the crystal unit, the oscillation frequency depends on the capacitive load that is applied. The amount of nonlinear frequency change as a function of load capacitance is referred to as the pullability. It indicates how far from the nominal frequency (intended oscillating frequency) the resonant frequency can be forced by applying the load [1]. Typically, it is used to tune the operating frequency to a desired value. In special cases, it can also be used for the measurement purposes, allowing the measurement of various quantities based on capacitive influence on the quartz crystal oscillation frequency [2-3]. However, when these various quantities are measured, the problem of insufficient sensitivity and nonlinear characteristics very often arises.

This research focuses on the pulling sensitivity and linearity of the AT fundamental quartz crystals (cut angle: +2′) operating over the measurement temperature range of 10 to 40 °C. Crystals fabricated in this manner exhibit excellent frequency vs. temperature stability. They have fundamental resonant frequencies between 3.5 and 21 MHz.

The equivalent circuit is an electrical representation of the quartz crystal's mechanical and electrical behaviour. The components C, L, and R are called the motional arm and represent the mechanical behaviour of the crystal element. C_o represents the electrical behaviour of the crystal element and holder. Typical quartz data of 3.5 MHz resonance frequency (fundamental mode) is as follows: fr = 3.5 MHz, C = 25 fF, L = 82.8 mH, R = 10 Ω and C_o = 4 pF. The values in the quartz crystal equivalent circuit were measured by a HP 4194A impedance/gain-phase analyzer.

The capacitance C_o is a real capacitance, comprising the capacitance between the electrodes and the stray capacitance associated with the mounting structure. It is also known as the “shunt” or “static” capacitance, and represents the crystal in a non-operational, or static state. Depending on the particular enclosure type, C_o normally lies between 1 and 7 pF. Oscillator crystals are normally designed with C_o less than 7 pF. One possibility how to increase the pulling sensitivity is to reduce C_o in Equation 1, which is the serial resonant frequency for the crystal in series with load capacitence C_z [1]: (1) fs ( Cz ) : = 1 + C Co + Cz 2 ⋅ π L ⋅ C

The other possibility is to compensate C_o with parallel inductance L_p connected to basic quartz crystal equivalent circuit providing that ω_o·L_p = 1/ (ω_o·C_o), resulting in Equation 2 [4-7]: (2) fsp ( Cz ) : = 1 + C Cz 2 ⋅ π L ⋅ C

The novelty lies in the compensation of C_o with inductance L_pw providing that ω_o·L_pw = 1/ (ω_o·C_o) (Figure 1) with the criterion k·L_pw = 1/ k·C_o. Resistance R_pw is a real part of impedance Z_pw.

Figure 2 shows a comparison of the pulling sensitivity between Equations 1 and 2 for a 3.5 MHz quartz crystal in the capacitance range of 3–50 pF. The change of frequency is approximately two times higher if we compensate C_o [dependence f_sp(C_z)]. For the general sensitivity measurement purposes, the capacitance range between 3 pF and 20 pF is the most useful. It is in this range that the frequency capacitance dependence is the greatest (the highest pulling sensitivity). The highest frequency sensitivity is in the range 3–10 pF, where a very small capacitance changes can be measured (aF range). Figure 2 also shows typical nonlinear frequency dependence and a small pulling range [dfs(Cz) ∼ 40 kHz].

Taking into account that: (3) fr : = 1 2 ⋅ π L ⋅ Cwhere -f_r is resonant frequency with phase 0, and (4) ω 0 : = 2 ⋅ π ⋅ f rusing the inductance L_pw connected in series and providing that k·L_pw = 1/ k·C_o (Equation 5), (5) Lpw : = 1 ω o 2 ⋅ Cowe get Equation 6, where we have a linearized frequency dependance with regard to C_z for various C_o values which change with “k” values. This represents a novelty in this research. Due to very small inductance L_pw, the real resistance R_pw can be ignored (Figure 1) [4].

k = 1, 2, 3 (6) fsk ( Cz ) : = 1 + C 2 ⋅ ( 1 k ⋅ Co − 1 ω 0 2 ⋅ k ⋅ Lpw − 1 Cz ) 2 ⋅ π L ⋅ C

Table 12 shows typical quartz data and experimental pulling results (dfs, dfsk for k = 1 and k = 2) for four different quartz crystals in the frequency range 3.5 to 21 MHz. The novelty here is in the pullability increase (dfsk) at values k = 1 in k = 2 (Equation 6), taking into account Equation 5. The values in the quartz crystal equivalent circuit were measured by the HP 4194A impedance/gain-phase analyzer.

For the given various frequencies the quartz crystal data could also be different (C₀, C, R, L). The frequency changes of dfs and dfsk (for k = 1 and 2) are measured at various C_z values (3 pF and 50 pF).

The maximum attainable stability of a crystal unit is dependent on the high Q value (3.5 MHz – Table 2). The smaller the distance between f_r (series resonant frequency) and f_p (parallel resonant frequency) the higher the Q value, and the steeper the slope of the reactance. The factors that further limit the Q are mounting loss, atmospheric loading (for non-evacuated crystal units) and the surface finish of the blank. Mounting loss depends upon the degree of trapping produced by the electrode and the plate diameter. The highest Q of quartz unit is important because of the frequency stability: (7) Q : = ω o ⋅ L R

The figure of merit is a useful indicator, particularly for oscillator application and shows as the difference between f_r and f_s (pulling frequency difference value). In an oscillator, for a resonator with M less than 2, the sustaining circuits must present inductive impedance to the crystal unit. As M increases beyond 2, f_r and f_a separate and, for large M, approach f_s and f_p, respectively. In general, the larger M is the more useful resonator (the greater crystal oscillation stability): (8) M : = 1 ω o ⋅ R ⋅ Co

As a consequence of hysteresis, the frequency vs. temperature curves obtained by slowly increasing the temperature from, say, 10 °C to 40 °C will not coincide with the curve obtained by slowly decreasing the temperature from 40 °C to 10 °C. Frequency stability also depends on the temperature coefficient of the core material used of L_pw. The proper choice of the core material is also the key in the sense of the frequency stability.

In general, the oscillator's circuit long-term stability also depends upon the crystal aging. Cold weld packages which are specially processed and welded in a high vacuum offer much better ageing rates and typically the ageing rates of cold weld crystal is less than ±1 ppm/year (10 °C to 40 °C). Stability of the electronic circuit depends upon the circuit type and quality of its elements [8-11]. The frequency stability ±0.1 Hz can be achieved provided that the above mentioned facts are taken into account and that the oscillator circuit is selected appropriately [1]. Another very important criterion for oscillator application is the drive level, which may not exceed 20 μW [12].

By satisfying the conditions of Equation 5, Equation 6 may be written, considerably increasing the frequency pulling range and linearizing it as shown in Figure 3 for the experimental data for the typical quartz data of 3.5 MHz resonance frequency (fundamental mode).

Figure 3 shows linear dependence for various k values as well as increased pulling (depending on the values of C_o and L_pw [Equation (6)] and also shows frequency dependences for serial resonant frequency fs(C_z) (without compensation), dependance of function fs1(C_z) (with compensation) for k = 1, fs2(C_z) for k = 2 and fs3(C_z) for k = 3, and the last one being very hard to achieve due to a very small capacitance C_o ≅ 1.33 pF. The oscillator frequency measurement error is approximately ±0.1 Hz [1].

We can define pulling sensitivity S as the frequency change in parts per million per pF change at a given load capacitance C_z for various k: (9) S : = C 2 ⋅ ( 1 k Co + Cz ) 2

In such a way, we can determine “S” for 3 to 50 pF. Since C_o and C are the same throughout our experiments (Table 1), we get the same “S” values for different k values (Table 3) for the frequencies 3.5 to 21 MHz (Table 2).

If we define the frequency ratio Ω = ω/ω₀, which depends on ω 0 = 1 / L ⋅ C, and taking into account ω₀L = 1/ω₀C, the impedance equation for a crystal unit with C_z and L_pw is [1], [13]:

Ω = 0.998, 0.99802…1.038.

At the frequency of 3.5 MHz and data for C_z = 3 × 10^-12 and k = 1, 2, 3 we get the three complex impedances as shown on Figure 4 (Equation 10). Ω, representing the change of resonant frequency in the vicinity of the serial resonant frequency f_r (the point where Im(Z_k) = 0 represents the series resonant frequency f_r). Figure 4 shows that by increasing the “k” values, Im(Z_k(Ω)) and the pulling sensitivity of the quartz crystal increase as well as illustrated on Figure 3 (at value k = 3 the complex value of impedance Im(Zk3(Ω)) is doubled).

Experimental results of the comparison between compensated quartz crystal equivalent circuit and those using a non-compensated quartz crystal equivalent circuit show that the use of series reactance compensated crystals of the same frequency increases the pulling range by ×25 for k = 2 and ×100 for k = 3 also depending on the circuit used. It is the increase of pulling and a simultaneous linearilization that represents a novelty and a major advantage of this method in the measurement of femto and atofarad ranges. When the load capacitance is connected in series with the crystal, the frequency of operation of the oscillator is linearly increased inside limited values. It should also be emphasized that the exact pulling limits depend on the crystal's Q-value as well as associated stray capacitances. The most common factors affecting frequency stability such as operating temperature range, aging, hysteresis and drive level as well as all other crystal characteristics influencing the stability should also be considered because a stable oscillator circuit plays an important role in the increase of pulling and linear frequency dependance. Increased pulling range obtained experimentally can be used for determination of many different measurements such as strain, compression, positioning, angle, level, pressure, humidity, dielectric, biological growth, bacteria growth, and many other non-electrical quantities [14-16].