Detection Principles of Temperature Compensated Oscillators with Reactance Influence on Piezoelectric Resonator

Abstract

This review presents various ways of detection of different physical quantities based on the frequency change of oscillators using piezoelectric crystals. These are influenced by the reactance changes modifying their electrical characteristics. Reactance in series, in parallel, or a combination of reactances can impact the electrical crystal substitute model by influencing its resonant oscillation frequency. In this way, various physical quantities near resonance can be detected with great sensitivity through a small change of capacitance or inductance. A piezoelectric crystal impedance circle and the mode of frequency changing around the resonant frequency change are shown. This review also presents the influence of reactance on the piezoelectric crystal, the way in which the capacitance lost among the crystal's electrodes is compensated, and how the mode of oscillators' output frequency is converted to lower frequency range (1-100 kHz). Finally, the review also explains the temperature-frequency compensation of the crystals' characteristics in oscillators that use temperature-frequency pair of crystals and the procedure of the compensation of crystals own temperature characteristics based on the method switching between the active and reference reactance. For the latter, the experimental results of the oscillator's output frequency stability (f_out = ±0.002 ppm) at dynamical change of environment temperature (0-50 °C) are shown.

Keywords: detection principle of piezoelectric oscillators; piezoelectric impedance; reactance influence on resonance.

Figure 1

The piezoelectric crystal equivalent circuit (Butterworth Van-Dyke (BVD) model) [19,20].

Figure 2

The complex impedance circle for the piezoelectric crystal equivalent circuit (C₁ = 3.24 fF, L₁ = 7281 H, R₁ = 30 kΩ). Both axes are in Ohm [22].

Figure 3

(a) Capacitance C_L influence (serial and parallel) on the piezoelectric equivalent circuit; (b) The reactance curves without a load capacitance C_L, with load capacitance C_L in series and then in parallel with piezoelectric resonator [21].

Figure 4

The range of resonant frequency changing $f_s{}^{*}$ = 32.7710 to 32.7745 kHz due to the changing of capacitance C_L in the range of 5–15 pF [18,21].

Figure 5

(a) Compensation C₀ with the inductance L_L connected in series; (b) compensation C₀ with parallel inductance L_p.

Figure 6

A comparison between the frequency changes $f_s{}^{*}\left(C_L\right)$ (Equation (12)), frequency $f_{ss}{}^{*}\left(C_L\right)$ (Equation (14)), and $f_{sp}{}^{*}\left(C_L\right)$ (Equation (16)) as result of changing of capacitance C_L (in the range from od 5–15 pF).

Figure 7

Load inductance L_LL in series with the piezoelectric crystal.

Figure 8

Inductance frequency pulling for data: C₁ = 3.24 fF, L₁ = 7281 H, R₁ = 30 kΩ, f₀ = 32.768 kHz, and for inductance L_LL change in the range 7.1–7.5 H.

Figure 9

(a) Higher oscillator frequency (f_osc) signal transformation to lower frequency range (f_out1 = 1–100 kHz), (NAND stands for digital AND gate with negated output, VCC = 5 V); (b) detection oscillator.

Figure 10

Temperature–frequency characteristics for thickness-shear mode AT-cut crystals with angle of cut from −4′ to +16′ as a parameter [6,21,32,35].

Figure 11

A temperature sensitivity of the synthetic crystals dependent on cutting angle [6,21,32,35].

Figure 12

Detection principle by two oscillators with similar crystal temperature–frequency characteristics.

Figure 13

(a) Detection principle by one oscillator and the switching mode method compensating crystal’s own temperature–frequency characteristics and crystal’s electrode capacitance; (b) experimental piezoelectric detection oscillator with two equal capacitances C_x and C_ref and oscillator symmetrical construction [16].

Figure 14

Detection principle by one oscillator and switching mode method compensating the crystal’s temperature–frequency characteristics and the crystal’s electrode capacitance by the same inductance.

Figure 15

Frequency stability $((f_{Lx}\left(t_2\right)+d{f}_{Lx}\left(T_2\right)+d{f}_{Lx}\left(t_2\right)-f_r+f_{c\_error\overline{Qt2}})$ occurring when changing the temperature in the range 0–50 °C (measurement time 2.5 h—two cycles).

Figure 16

Frequency difference stability $d{f}_{out4}\left(Q,\overline{Q},t\right)$ at the change of temperature in the range 0–50 °C, and fixed value L_x (measurement time 2.5 h—two cycles).