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This paper presents a new method of substantially improving frequency pullability and linearity using reactance in series with an AT fundamental crystal operated with a series load capacitance in the range of 3 to 50 pF and frequencies in the range of 3.5 to 21 MHz. The research describes high quartz pullability and linearity by varying the load capacitance. The paper also gives impedance circuits for crystal unit (3.5 MHz) together with load capacitance and compensation reactance. The experimental results show that the new approach using compensation method of quartz crystal connected in series reactance increases the frequency pulling range by ×25 to ×100 depending on the type of oscillator and compensation factor “k” in the temperature range of 10 to 40 °C.

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 [

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

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 _{z} [

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

The novelty lies in the compensation of C_{o} with inductance L_{pw} providing that ω_{o}·L_{pw} = 1/ (ω_{o}·C_{o}) (_{pw} = 1/ k·C_{o}. Resistance R_{pw} is a real part of impedance Z_{pw}.

_{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).

Taking into account that:
_{r} is resonant frequency with phase 0, and
_{pw} connected in series and providing that k·L_{pw} = 1/ k·C_{o} (_{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 (

k = 1, 2, 3

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 (

For the given various frequencies the quartz crystal data could also be different (C_{0}, 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 – _{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:

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):

As a consequence of hysteresis, the frequency _{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 [

By satisfying the conditions of

_{o} and L_{pw} [_{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 [

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:

In such a way, we can determine “S” for 3 to 50 pF. Since C_{o} and C are the same throughout our experiments (

If we define the frequency ratio Ω = ω/ω_{0}, which depends on
_{0}L = 1/ω_{0}C, the impedance equation for a crystal unit with C_{z} and L_{pw} is [

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 _{r} (the point where Im(Z_{k}) = 0 represents the series resonant frequency f_{r}). _{k}(Ω)) and the pulling sensitivity of the quartz crystal increase as well as illustrated on

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 [

Load capacitance C_{z} and compensation impedance Z_{pw} = j·ω·L_{pw} + R_{pw} in series with the quartz crystal equivalent circuit.

Quartz crystal pulling sensitivity from 3 to 50 pF.

Quartz crystal pulling and linearization for k = 1, 2, 3 in the range C_{z} = 3–50 pF.

Compensated quartz impedance (C_{o}) for different k = 1, 2, 3 (Ω = 0.998, 0.99802…1.038).

Quartz data and pulling sensitivity in frequency range 3.5 MHz to 21 MHz.

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Pulling sensitivity measured between C_{z} = 3 and 50 pF | ||||||||||

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k = 1 | k = 2 | |||||||||

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f_{r} (MHz) |
R(Ohm) | C(fF) | L(mH) | C_{o}(pF) |
L_{pw}(uH) |
Q (k) | M | dfs(kHz) | dfsk(kHz) | dfsk(MHz) |

3.5 | 10 | 25 | 82.83 | 4 | 520.0 | 181.98 | 1,137 | 5,431 | 128 | 0.514 |

9 | 10 | 25 | 12.53 | 4 | 78.2 | 70.68 | 442 | 13,979 | 330 | 1,322 |

15 | 10 | 25 | 4.46 | 4 | 28.1 | 42.03 | 266 | 23,402 | 553 | 2,214 |

21 | 10 | 25 | 2.30 | 4 | 14.4 | 30.34 | 188 | 32,589 | 771 | 3,083 |

Quartz data and pulling sensitivity S.

_{z} = 3 pF |
_{z} = 50 pF | |||||
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k = 1 | k = 2 | k = 3 | k = 1 | k = 2 | k = 3 | |

S | 2.551 × 10^{8} |
5 × 10^{8} |
6.657 × 10^{8} |
4.287 × 10^{8} |
4.663 × 10^{8} |
4.744 × 10^{8} |