A multimode-quartz-crystal oscillator was developed to excite stable dual-mode resonance at different frequencies: The oscillation of the 3rd harmonic resonance of the principle C-mode and an additional resonance B-mode of SC-cut crystal. Harmonic combinations of the 3rd and fundamental mode of B-mode with the 3rd harmonics of C-mode are demonstrated. The measurement of the temperature dependence of the oscillation frequency is demonstrated along with the stability determined by root Allan variance. Dependence on the open conductance of the active circuit and the dependence on the coupling capacitors are discussed.
Piezoelectric sensing is increasingly attracting attentions, employing frequency-based measurement depending on the surface loading, mass, and a variety of force acting on the piezoelectric resonator. The quartz crystal resonator determines a stable electrical resonance frequency corresponding to the mechanical resonance frequency. The quartz crystal resonator shows a variety of resonance modes. For example, in an AT-cut quartz crystal, the harmonic combination of the different temperature characteristics of the fundamental, 3rd and 5th harmonic resonances of provides a high precision thermometric system. In the past history, dual-mode oscillation was attempted by several groups and is reviewed in References [
Local resonance circuits consist of L02 and C0x, L12 and C1x. C0y and C1y are inserted to the connection to a crystal resonator. Feedback resistors R02, R03, R12, and R13 are inserted into the power line between CMOS inverter integrated circuits IC1 and IC2 and ground aimed at suppression of current and gain control. C05, C06, C15, and C16 are pass capacitors. C02, C03, C12, and C13 are necessary for the generation of negative resistance. This circuit synthesizes LC oscillation mode determined by L02 and combined capacitors C0x, C02, C03, and C0y, and LC oscillation mode determined by L12 and combined capacitors C1x, C12, C13, and C1y. The oscillation frequency is settled in the vicinity of the quartz crystal oscillation frequency. Coupling capacitors C04 and C14 are omitted in the following analysis. Coupling capacitors C0y and C1y are connected to the quartz crystal resonator. Appropriate choice of the local resonance circuit and the coupling capacitors are necessary to realize stable dual-mode quartz crystal oscillation. Two oscillator circuits OSC1 and OSC2 are connected to the resonator by capacitors C0y and C1y. The SC-cut quartz crystal resonator shows C-mode a primary oscillation mode with good stability, and B-mode, a side mode, linearly proportional to the change of ambient temperature. CMOS Inverters IC1 and IC2 are replaced with a constant current source. In the equivalent circuit on the left side is OSC1 and on the right side is OSC2. Iout1 and Iout2 are output current, and Vin1 and Vin2 are input voltage of each inverter: r02 and r12 are internal resistance of inductance L02 and L12, respectively. C0s and C1s are stray capacitance parallel to capacitance C0x and C1s. Impedance Z01 consists of L02, r02, C0x, and C0s, similarly, impedance of the circuit Z11 which consists of L12, r12, C1x, and C1s, and Zxt for the parallel circuit of a quartz resonator and Cp, reduced impedance is found. Capacitance of leads and connected lines on the circuit board are modeled by Cp. Applying Kirchhoff’s law, homogeneous Equation (1) is found.
The open conductance Gm is adjusted by negative feedback resistors R02, R03, R12, and R13. Equivalent trans- conductance is defined by the open conductance and a feedback resistance, as in (2).
The open conductance of the CMOS inverter is expressed with drain current Id and conductance coefficient K defined in the terms of electron and hole mobility μ and the unit gap capacitance Cg. W/L is the ratio of width over length of the gate:
Feedback resistance:
Solving the determinant of the coefficient matrix, relation (3) is found. The second term on the left side of this relation indicates the impedance of OSC1, and the third term indicates the impedance of OSC2. The impedance of individual oscillator is connected in parallel to Zxt.
ance. R1, C1, Ra, La, C0xs, ω0x, ω0s, and ω1 are found in (4). R2, C2, Rb, Lb, C1xs, ω1x, ω1s, and ω2 are found in (5).
Similarly,
where
Including the equivalent circuit of Zxt, equivalent circuit-3 is found in
are found.
where
Including C0, Cp and Cz into Cci and Rci composed equivalent impedance Ccci and Rcci are found, as in (7).
Supposing i(ω) in the closed circuit of this diagram, the resistance condition is found from the matching condition for the real part where the real part is equal to zero. Oscillation frequency is determined from the matching condition of the capacitance, where the imaginary part is equal to zero.
Oscillation conditions are given as the resistance condition and frequency condition, in the following forms (8) and (9), respectively:
Mode | Equivalent Circuit Constant | |||||
---|---|---|---|---|---|---|
C0 (pF) | L1 (mH) | C1 (fF) | R1 (Ω) | f1 (MHz) | fr (MHz) | |
C-mode, 3rd harmonics | 3.431 | 1297 | 0.1954 | 70.5 | 9.997418 | 9.999807 |
B-mode 3rd harmonics | 3.244 | 1757 | 0.12 | 264 | 10.96083 | 10.952481 |
B-mode fundamental | 3.899 | 495.3 | 3.710 | 61.8 | 3.712778 | 3.712836 |
Inequality in condition (8) indicates that the negative resistance larger than damping is necessary at the beginning of oscillation. The series resonance frequency f1 is determined by L1 and C1 determined in relation (10), where ω1 is angular frequency and fr is the series resonance frequency.
The resistance condition is estimated for the oscillation of each mode from
In the case of the fundamental resonance of B-mode, the series resistance is reduced to 61.77Ω. Negative resistance of the active circuit is indicated by the absolute value. In the physical meaning, this value balances with the damping; this value is expressed in the following form:
The frequency dependence of negative resistance is tailored considering the influence of resonator capacitors C0x and C1x. The oscillation condition of the 3rd harmonics of C-mode is fulfilled at C1x = 20.5 pF, where the oscillation frequency is close to the series resonance frequency. The maximum absolute value of negative resistance is approximately 1.99 kΩ, at 9.992 MHz. The frequency range varies from 0.28 to 0.3 MHz. Similarly, the oscillation condition of the 3rd harmonics B-mode is fulfilled at C0x = 13.5 pF, close to the series resonance frequency f1b. The maximum absolute value of negative resistance is approximately 1.9 kΩ, at 10.9 MHz. The oscillation condition is fulfilled from 10.84 to 10.96 MHz. The frequency range varies from 0.11 to 0.12 MHz. In the combination of the fundamental oscillation of B-mode and the 3rd harmonic resonance of C-mode, the dual mode oscillation is realized at C0x = 31 pF and C1x = 20.5 pF.
In the combination of the 3rd harmonic resonance modes, the multimode-oscillation is observed provided that the oscillation condition is fulfilled simultaneously. In the stable oscillation, gain G1M decreases compared with
the initial phase. At G1M = 1 mA/V, the damping resistance becomes larger than negative resistance. The maximum absolute value of negative resistance is approximately equal to 2 kΩ and the damping resistance of the 3rd harmonic resonance of B-mode is 264 Ω. The negative resistance of the oscillator is shown for various values of gain G1M, in the vicinity of 10.9 MHz and 10 MHz.
In
The gain at the initial stage of oscillation is set at 5 mA/V, along with the growth of the signal, B-mode shows a maximum value of negative resistance 4.5 kΩ at G1M of approximately 2 mA/V, then steeply decreases. In
70.5 Ω the oscillation condition of C-mode is fulfilled before the oscillation condition of B-mode is achieved, if negative resistance is smaller than 264 Ω. In this case, the oscillation mode transfers to the single oscillation of C-mode.
In
fixed at 10 pF. The case of C0y = 0 pF means that the oscillator is disconnected from the quartz resonator and only OSC1 is connected to the resonator. The oscillation condition for B-mode (OSC1) is fulfilled in a narrow region in the vicinity of the resonance frequency, in the case where C1y = 30 pF:
In
dition for C-mode is fulfilled from 9.8 to 10.1 MHz and for B-mode from 10.8 to 10.9 MHz. If C0y is set to 30 pF, the oscillation condition for OSC2 is fulfilled, but the negative resistance for OSC1 is lower than the critical value.
On the other hand, negative resistance of the B-mode oscillator fulfills the oscillation condition only in a narrower region from 6 to 18 pF.
The oscillation condition of C-mode is fulfilled, at C0y and C1y from 2 to 48 pF, and stable oscillation of C-mode is available. The oscillation condition of B-mode is fulfilled only at narrower region of C0y and C1y from 6 to 13 pF. Stable dual mode resonance oscillation is available in this region.
The short-time stability of the C-mode oscillation is compared. The bottom level of the σ versus τ dependence is employed as the measure of the stability, where σ is the root Allan variance defined as (12) and τ is the sampling interval (gate time) [
fk is the moving average of 10 sequential of frequency data. τ is the sampling time interval. M is the number of samples per measurement. The average of frequency is calculated over the gate time. Measurement was carried out by universal frequency counters Agilent 53230A (Agilent Technologies, Santa Clara, Ca, USA) synchronized with a rubidium oscillator.
The stability of the single C-mode oscillation is compared with the simultaneous multimode oscillation of different harmonic combinations. In this experiment, the measure of the stability is the bottom level of the σ versus τ dependence, where σ is the root Allan variance defined as (12) and τ is the sampling interval (gate time).
Temperature dependence of the oscillation frequency was measured in the simultaneous oscillation. Allan standard deviation was observed to avoid spontaneous mode change of additional resonance mode.
The regression curve of C-mode shows a cubic function with an inflection point at 85.7˚C (Designed value). The frequency drift of C-mode is 0.32 × 10−6 per degree Celsius. This result suggests that the stability of the oscillator remains at the level of typical quartz crystal oscillator. The regression curve of B-mode indicates a linear dependence of approximately −26.5 ppm/degree. The dual-mode oscillation direct thermometry is applicable over this temperature range.
The common part of circuit constants for the analysis are as follows: Oscillator 1: C02 and C03 = 47 pF; C0s = 5 pF; C0y = 10 pF; G1M = 5 mA/V. Circuit constant of Oscillator 2: C12 and C13 = 47 pF; C1s = 5 pF; C1y = 10 pF; G2M = 5 mA/V; Cz = 100 pF; Cp = 2 pF; C0c = 3.431 pF; C0b = 3.244 pF. L02 = 5.6 μH; L12 = 5.6 μH; C0x = 9 pF; C1x = 17 pF; L02 = 33 μH for the fundamental mode or L02 = 5.6 μH for the 3rd harmonic resonance of B-mode.
The common part of circuit constants for this experimental part is as follows. Oscillator 1: C02 and C03 = 47 pF; C0y = 10 pF; L02 = 5.6 μH; C0x = 9 pF for the 3rd harmonic resonance and L02 = 33 μH; C0x = 30 pF for the
fundamental resonance of B-mode. Oscillator 2: C12 and C13 = 47 pF; C1y = 10 pF; L12 = 5.6 μH; C1x = 17 pF for the 3rd harmonic resonance of C-mode. IC1 and IC2 TC7SHU04F, Vcc = 5 V.
The SC-cut resonator is made of double-rotated cut crystals and it shows a principal resonance C-mode and B-mode, additional resonance mode. High stability of the principal mode was observed in both single and multimode resonance oscillation. The linear dependence of the B-mode on the temperature can realized the thermometric method with the assurances of the stability. Design rules were ensured by the measurement of the stability of σ versus τ curve.
The narrow-band wide variable crystal oscillator circuit was proved to be an efficient experimental maneuver in the simultaneous measurement of the stability and the oscillation frequency dependence on the ambient temperature. The frequency separation is narrower than the combination of the 3rd harmonic resonance of C-mode and the fundamental resonance of B-mode which is approximately 6.2 MHz. The combination of the 3rd harmonic resonance oscillation shows quasi-sinusoidal wave forms with negligible distortion with Allan standard deviation σy(τ), 10−11 < σy(τ) < 10−10, sufficiently stable for standard time base and sensor applications. In the present stage of the experiment, the demonstrated result was obtained by precise choice of the circuit constant and bias current. It is necessary to monitor the unlocking from crystal resonance by the real-time monitoring of the Allan standard deviation. Although the reproducibility of the circuit and the oscillation condition is not thoroughly identified and appropriate circuit design is still necessary, we risk concluding that the dual-mode quartz crystal resonance oscillation can be applied to piezoelectric sensing and the production of a stable frequency standard.
We express acknowledgement to Ms Ruzaini Izyan binti Ruslan and Ms Izzati Syazwani binti Othman for their collaboration in the early stage of experiment, and Mr. Masahide Marumo, River Eletec Company (Japan) for fruitful discussions.