Low-frequency double-resonance quartz crystal oscillator was developed with active inductance circuit aiming the start-up of stable oscillation of tuning fork-type quartz crystal resonator at 32.768 kHz within 0.37 ms. The initial oscillation is triggered by a part of crystal oscillator forming a CR oscillator. The negative resistance ranges to 4 MΩ at g mf of 4.1 μA/V. In a limited frequency range, the circuit shows negative reactance C cci = -3.4 pF equivalent to inductance L cc = 9.8 H. The Allan standard deviation indicated 10 -11 to 10 -10, showing high stability comparable to general quartz crystal oscillator.
Piezoelectric quartz crystal oscillators have widely expanded in sensing of the environmental data such as static pressure and temperature. Acceleration of the piezoelectric oscillator enables the intermittent operation of the piezoelectric sensor for power management. Engineering issue in the acceleration of the start-up of low frequency quartz crystal oscillator includes 1) triggering circuit, 2) generation of large negative resistance, and 3) linearity of the active device in large amplitude oscillation. In this work, we aim at the acceleration of the start- up of low frequency, tuning fork quartz crystal resonator within several oscillation periods, which enables the intermittent operation of the sensor system. Acceleration of the start-up is studied by the gain control in the quartz crystal oscillator using a cascade circuit in the frequency region of several Mega Hertz [
Applying Kirchhoff’s law, the relations for Iout and Vin are found. Vin is the input voltage of IC1 and Iout is the output current of IC2.
Solving for the relation between Iout and Vin, total conductance GM is found.
Then the following relation is found. Current I2, I3 are expressed in the terms of I1.
Rearranging the expression, relation (11) is found.
Z2 is the impedance of a quartz crystal resonator (Zxt), and impedance for other components is defined as in (12). The composed impedance Zcc of the active circuit is found, substituting the impedance. From the condition for the non-zero solution of current, the oscillation condition results in (13). The impedance of the circuit is divided into resistive and reactance parts.
The equivalent resistance and the reactance of the circuit are found. Equivalent inductance Lcc or capacitance Ccc is determined depending on sign of reactance Xcc.
Factors “a”, “b”, “c” and “d” are introduced for the simplicity of the expression, where factors “c” and “d” have the dimension of Ω and factors “a” and “b” are dimensionless numbers.
GM is separated into real and imaginary parts.
Introducing (13) and (19) into Zcc, the impedance of the active circuit is found.
The active circuit is indicated with Rcc and reactance Ccc or Lcc depending on the sign. The resonator consists of parallel capacitance C0 and the motion arm, L1, C1, and R1, the equivalent series inductor, capacitor, and resistor respectively. CS is a stray capacitance. Calculating the parallel composition of C0 and Cs with the active circuit, equivalent circuit-3 in
Negatively signed capacitance is converted to an active inductance by relation (21),
The denominator of negative resistance Rcci has quadratic dependence on Rcc. The maximum value of the absolute value is reached at a specific value of Rcc determined by C0s and Ccc. The following relation is fulfilled.
The active inductance appears in the vicinity of the resonance frequency, while capacitance Ccc is negative. The resonance frequency is determined by Lcc, C0s, and the sum of C0 and Cs. In this simplified form, the absolute value of negative resistance Rcci becomes infinitely large, if Ccc approaches-C0s and condition (23) is fulfilled.
At the resonance frequency determined by Lcc and C0s, the absolute value of negative resistance determines the growth of signal. The suppression of negative resistance by inductance L1 establishes the stability and inhibitory action against the signal growth.
Quartz resonator | τ(ms) | ||||
---|---|---|---|---|---|
L1 | C1 | R1 | C0 | Q1 | |
Tuning fork-type (32.768 kHz) | 11,797 H | 2 fF | 47.6 kΩ | 1.14 pF | 51,023 |
inductance Lcx = 12 H.
In
This result suggests a design principle of the circuit: Higher resonance frequency needs higher gmf. In this analysis, we take a look at the circuit impedance from the resonator terminal. The parallel capacitance C0 and stray capacitance Cs are included in the impedance of the active circuit. Thus, the relation between Rcci and Rcc is presented, As a part of the final solution, the result that Rcci becomes infinitely large at Ccc = −C0s, must be interpreted carefully in the context of the actual circuit design. The minimum idea given here is that the active inductance can generate large negative resistance compared to the capacitive region. Actually, Rcc is determined by number of circuit constants and angular frequency of the oscillation, and the strength of the oscillation is limited within the linear region of the active circuit.
The crystal current through the motion arm is not generated in the initial stage of the oscillation. In another
expression, this branch does not exists in the circuit. Because of high Q, the start-up needs reasonable acceleration system.
The start-up mode of the oscillator depends on the rise of Vcc. When the bias current increases the CR oscillation as in Mode-1 starts before the establishment of the crystal current. The oscillation frequency of Mode-1 is determined by R2 multiplied by the composed capacitance. When the quartz crystal resonator is activated sufficiently, the motion arm appears in the circuit, as in Model-2. Computer simulation was carried out using LT- spice for Windows (Linear Technology Corporation, 1630 McCarthy Blvd., Milpitas, CA, USA) [
The circuit constants of the motion arm are not corresponding to the values assigned in the analysis and experiment. Also, the delayed connection of the motion arm is not considered in this simulation.
When the motion arm is removed, this circuit forms a CR oscillator. The oscillation frequency is determined by the reactance of the parallel capacitance of the quartz resonator and feedback resistor R2.
The start-up and the stability of the stable oscillation of the double resonance oscillator is experimentally evaluated. The stability of the oscillation frequency is analyzed with 53,230 A universal frequency counter (Agilent Technologies, Santa Clara, Ca, USA) synchronized with external rubidium oscillator with long period stability < 2 × 10−11/month and short period stability < 1 × 10−11/s.
The proposed quartz oscillator circuit is activated with Vcc and the minimum start-up time marked 0.37 ms, as shown in
In this experiment, modified two-sample Allan standard deviation is employed as a measure of the short-time frequency stability. This protocol is defined in (25), following IEEE Standard 1139 [
deviation of 10−11 satisfy the requirement for the standard sensing. Probably, for the further improvement of the stability, the improvement of the Q-value of the resonator is necessary.
Quartz oscillator circuit | τ(ms) | |||
---|---|---|---|---|
1 | 10 | 100 | 1000 | |
Double-Resonance Quartz Oscillator | 2.7 × 10−7 | 2.4 × 10−8 | 2.7 × 10−10 | 6.8 × 10−11 |
Environmental sensing awaits solutions to reduce the electric-power, in continuous monitoring. The quick start of quartz crystal oscillators allows excitation of stationary oscillation established after short transient meeting the request for the power management in the environmental sensing such as the pressure and temperature. In this work, we resolved the engineering issues for the rapid start-up: 1) Large negative resistance; 2) Low distortion and linearity; 3) Triggering circuit. The start-up of a low frequency quartz oscillator is triggered with a CR oscillator and transferred to a quartz crystal oscillator. The maximum negative resistance ranges to 4 MΩ at specified gain of the active CMOS inverter circuit gmf = 4.1 μA/V. The composed reactance of the active circuit Ccci shows negative value, −3.4 pF which acts as inductance of 9.8 H and generates large negative resistance. Rapid start-up of the oscillation was established by the energy transfer by the initial CR oscillation of the active circuit and the minimum start-up time was realized. The oscillation condition was examined by the analysis and the start-up in the initial stage was examined by the computer simulation and experiment. The result shows corresponding dependence of the start-up time on circuit parameter R2. The stability performance of the double-re- sonance oscillator showed that short range stability of 10−11 satisfies the industrial requirement for the standard quartz oscillator circuit.
The authors acknowledge Ms Ruzaini Izyan binti Ruslan and Mr. Satoshi Goto for their collaboration in the early stage of this experiment. This work was supported in part by JST A-STEP Contract No. AS251Z01794J.